Deputize the citizenry.

http://mayberry.wikia.com/wiki/Goodbye,_Sheriff_Taylor

A police officer can deputize citizens for extra help.

http://thelegalgeeks.com/2015/08/11/bat-jim-is-the-hero-we-deserve/

Private persons may assist law-enforcement officers in effecting
arrests and preventing escapes from custody when requested to do so by
the officer. When so requested, a private person has the same
authority to effect an arrest or prevent escape from custody as the
officer making the request. He does not incur civil or criminal
liability for an invalid arrest unless he knows the arrest to be
invalid. Nothing in this subsection constitutes justification for
willful, malicious or criminally negligent conduct by such person
which injures or endangers any person or property, nor shall it be
construed to excuse or justify the use of unreasonable or excessive
force.

In your world, if a law and order matter requires firearms, there is fortunately a large body of armed private citizens that the unarmed officer can deputize to help in the matter. Carrying a weapon means consenting to be deputized to use it in time of civic need.

If you deputize a number of individuals this might be equivalent to raising a militia.

In a "Switzerland like nation", there might be a cultural way of explaining this.

In the real Switzerland, there is a universal draft for all male citizens. If you are not a conscientious objector (can't remember the proper term in Switzerland), then you receive standard military training and are issued an automatic rifle. Once you complete your term of service, you are released to the Reserve and take your automatic rifle home with you. I'm not clear if this has changed, but it also used to be common for each citizen solder to have 200 rounds of ammunition at home. They were encouraged to go to the local range and practice, and could purchase replacement rounds at a low cost to keep their stockpile at home to 200 rounds.

The Swiss Citizen Militia means the population has an almost 100% availability of automatic firearms among the population (even the United States has nowhere near that availability of automatic firearms, so called "assault weapons" is a scare term to describe semi automatic firearms derived from AR-15 platforms), yet some of the lowest rates of gun crime in the world. This is largely because each citizen has been carefully instructed and drilled in the proper use of firearms through military service.

When the United States was more rural in nature (really up until the 1950's), most homes had firearms of some sort, hunting rifles or shotguns. Proper use of firearms was a family responsibility, with fathers, uncles and grandparents teaching their children and minor family members (usually through taking them hunting). In fact, this can be found in other places and times, if you read Conan-Doyle's "Sherlock Holmes" stories, Watson is routinely armed, and there are some occasions where they actually access firearms or armed help from other people in London, England. Try doing that now and it will be a different story.

So an armed population does not necessarily equate to a dangerous one, so long as firearms owners are trained and educated in the proper use of firearms. The police in Switzerland are not alarmed in any way that any house they visit has an automatic rifle (they have one too), because they have a virtual certainty that the owner is a responsible citizen.

The country could be policed just like any other country.

Police: Hey you, stop right there!

Pedestrian: What is the problem, sir?

Police: That was an illegal jaywalk you just performed. Do you have a
license to perform illegal jaywalks?

Pedestrian: Why yes, sir, I do. Here you go, have a look at this.

Police: This license is expired! You are under arrest for "jaywalking
with an invalid license." Put your hands behind your back.

Pedestrian does so.

Police cuffs and jails the heinous criminal.

Just like that.

A high percentage of gun ownership among the populace does not change anything that just happened. Most of the time, the firearm possessed by police does not even enter the equation and has no part in an event.

Potential problems?

Now let us examine the other extreme, where someone might think this could actually be a problem.

Assume a violent criminal possesses a firearm and that the law enforcement officer does not. This is the scenario in which trouble could arise. But think about that statement for a moment... "a violent criminal possesses a firearm." If this person is a violent criminal, then the fact that firearms are entirely legal is irrelevant. This person very well might have a firearm even if they were not legal.

So the exchange which could be problematic is not unique to a country where a pedestrian has a gun but police do not. In fact, this situation actually happens in reality. I recall an event, in London if I recall, a few years back in which two criminals were armed with a knife and a gun, and the local police, being unarmed, could do nothing but shout at them until the armed police unit arrived. The criminals were free to shoot at everyone until that time.

Potential benefits?

Now let's look at this from a different angle, about the possible benefit.

In the situation I mentioned in London, if a nearby pedestrian was armed with a gun, they could have helped the defenseless police instead of waiting and risking more lives. Or, if it is illegal in your hypothetical country for a pedestrian to do this, they could hand their gun over to the unarmed police to use in this life threatening situation, expecting it back after.

In fact, armed civilians have helped police in the past. There are instances of criminals attacking police and having them pinned down, where some nearby pedestrian has shot the criminal and saved the police.

Also, in a famous case where a pair of robbers wearing heavy full body armor were in a shootout with police, the police shots were not harming the criminals because of their heavy body armor. The police needed something better, and a local gun shop nearby handed over higher powered weapons for the police to use in the fight.

Also, in countries where firearms are illegal, the crime rates are generally not lower. Some people claim otherwise, pointing to gun-specific crimes being down, but the violent crime rates overall are generally not lower. In some such places, the crime rates are even higher, and interviewed criminals in prison have stated that they feel safer committing their crimes in those areas because they know the populace are not armed.

Conclusion

So how could they police the country? Well, they could allow civilians to take part in the policing. Or there could be a law that anyone who is armed and not in immediate danger must surrender their weapon to the police to use against a nearby threat (or it could just be a voluntary thing).

This could be a good thing as well, as it could help to reduce abuse by police. In most situations, the police are armed and those they are interacting with are not, so the police are able to easily bully people. This happens often. If you get too far out of line, they will draw their weapons. If they get too far out of line and you respond in your defense, they will draw their weapons. In your situation, people would be more free from the threat of police violence.

If you have a responsible populace, similar to Switzerland, this will likely be a more peaceful and safe place to live than what most of us are used to.

Some points to consider

Alexander asks,

"The problem here is not the legality of firearms, it's abundance of
them. In day-to-day operations, police has to react to a number of
incidents. Some of those incidents are involving guns. If number of
those incidents are high (more than armed policemen can handle),
should police dispatch unarmed officers to deal with them?"

That is a good question.

That depends on what you mean by "involving guns." If guns are being fired at people, then no, sending unarmed officers is just dumb. If by "involving guns" you mean "Police are responding to a non-violent crime, and the suspect just happens to have a pistol at their side or a rifle over their back," assuming you have a mature population, then yes, go ahead and send unarmed officers. The suspect is armed. So what? Police deal with armed suspects all the time without even knowing it in US states where open carry is illegal and everyone carries concealed; that has not been problematic.

Based on comments, I think there might be some confusion about my stance.

Obviously, it would be dumb to enter a dangerous situation while unprepared for it. I am often the first to say that we should be prepared for anything. I do not think it is wise to force the general police to be unarmed. The question states that there are unarmed police, so I am answering from that reality.

What I said is only meant to apply to people who have not been identified as violent. If you have any reason at all to believe that there could be violence, then yes it would be dumb to be unarmed. But the mere presence of a gun is not reason to believe that there could be violence. If the jaywalker in my answer's example had an AK-47 over their back but acted friendly, there is no reason to assume the worst. If they are known violent, or acting aggressive, or performed some worse crime that tends to lead to violence, I'd be concerned.

Similarly, if I was in Switzerland I would not feel in danger responding to a call where the suspect was armed but not deemed violent; but if I was in Mexico or certain Middle East areas or a known dangerous section of a major metropolitan area, I would feel in danger responding to any call, whether I knew a gun was present or not and I would not go unarmed.

Common sense is necessary. One of my points is just that the mere presence of a gun is not, in and of itself, cause for alarm. The appropriate response would be situational.

This is the non-USA reality (more or less)

Aaron has pretty much nailed it in his answer. Looking at Australia - an incident in which a police officer draws a firearm on duty is highly likely to make the national news and will definitely involve paperwork for the officer involved. An incident in which anyone is actually shot by the police (fatally or otherwise) always make national headlines. Given how often they are used, Australian police firearms are a psychological rather than a physical weapon.

However, there are relatively few incidences of firearms violence, despite:

• the average beat cop being a mediocre marksmanship (trust me - I used to compete against them in inter-services pistol competitions)


• the vast majority of privately owned firearms in Australia are rifles or shotguns, meaning that if firearms ever are involved the average beat cop is outgunned and definitely outranged

While the laws regarding gun ownership became relatively draconian following the Port Arthur massacre in 1996, private firearm ownership (both legal and illegal) continues. The key differences with the United States are:

• Firearms are socially unacceptable in most locales. No one openly carries firearms in public except for police and security guards for armoured cars. (Most private security are not allowed to carry firearms.) Even the military keep their weapons out of sight most of the time outside of training areas.


• Firearms ownership laws are not the same as firearms carriage laws. Laws can allow the populace to own rifles, but this does not stop laws requiring those rifles to be securely locked in a safe while stored or in a locked container while in transit and only brought out for cleaning or actual use on a designated range or permitted hunting area.


• Massive police response. As stated above, most civilians with firearms (legal or illegal) will have firepower that outguns what the poorly armed, indifferently trained beat cops or (mostly) unarmed security guards have available. So police and private security training is focused on calling for backup immediately. If the threat cannot be talked down immediately, this means that almost any firearm threat will result in a special response group being called out. In extreme situations (terrorism or something that looks like it) the police can call on special forces - no constitutional rules against that in Australia or most other countries. The criminals know that open possession of firearms will draw down more heat than they want, so they generally avoid it.


• Police culture. Australian police officers do not go for their guns except in extremis. As Sasha noted, police are trained to de-escalate situations by talking people down. If force is necessary it is far easier to justify use of pepper spray or a baton at the inevitable subsequent enquiry. (Australian police are not all saints and there are abuses, but firearm involvement in those is very rare.)

In fact one positive aspect for police in the proposed country with widespread private firearms ownership would hopefully be a more educated populace. The majority of Australians have knowledge of firearms based on what they see in Hollywood productions, leading to ridiculous criticisms when police do use firearms. ("Why didn't the policeman shoot the gun out of his hand instead of killing him?" "Why did they shoot someone who only had a knife charging them at short range?") Better education would hopefully alleviate at least some ignorance in the media and their audience.

In summary - policing a country with widespread firearms ownership is quite feasible with low levels of police armament with the right cultural foundation.

Assuming you're looking for a fictional answer, here's one from a near future.

Assume:

• Everyone well-identified (RFID or equivalent, plus facial recognition and biometrics)


• Internet everywhere (perpetually tracking the location of every identified person)


• A cashless society; with transactions permitted/authenticated based on "who you are" (and not so much on "what you have" -- so that unlike cash and so on it can't be stolen)

The penalties for being a criminal are then higher, IMO, almost insurmountable:

• Easy to show that it was you who done it


• A court order or arrest warrant could cut you off from any and all social commerce (shopping, food, lodging, public and private buildings and transportation vehicles) -- and dispatch a SWAT team to your location whenever you show up on the grid, if you were a fugitive

This could be especially so if you were a violent criminal (using a "gun" to resist law enforcement).

An outlaw would have to, I don't know, maybe, live in the woods like Robin Hood or some kind of cave man -- that's not really feasible IMO, especially given modern surveillance tech.

Another option (is this one even more utopic?) might be to ensure that the entire populace is wealthy, healthy, fairly happy, well-educated, fairly drug-free, treated for mental illness, and civic-minded (is that Switzerland again?), maybe they'd be mostly law-abiding of their own accord.

Hurry up with self-driving cars. Nearly half of the criminal convictions in Switzerland appears to be for "Major Violation of Traffic Laws"; if people stopped driving maybe that problem would go away.

I don't see how such a situation would arise in the first place.

If you look at the world ranking of estimated civilian guns per capita, you'll note that it's the United States and then everyone else. Switzerland, incidentally, which is frequently brought up has a lower rate than Canada, which some people (ie, Americans) overestimate when it comes to gun control. Anyway, if you're talking about a modern country, the majority of people would live in cities and quite honestly owning a gun in an urban is more bother than it's worth, especially if, as stated, a gun ban was recently lifted. Some people may rush out to get a firearm just because, but the majority of people in cities would not. So it's difficult to see how the ownership number would get so high.

Moreover, there is no way in hell there'd be freedom to own everything save explosives. Again, the United States (or at least some components of it) are very much the exception in the breadth of firearms allowed on the civilian market. For essentially everyone else there'd be restrictions on firearms in place, and I don't see how your country would be any different.

During the preceding years, the populace became overwhelmingly law-abiding, which means that the police also became overwhelmingly law-abiding (because getting the former without the latter is a trick nobody has been able to pull off).

So now even though anyone who took it into their minds could mow down the first few cops who tried to restrain him, very few people will.

I cannot point you to a reference—and therefore this tale may be fictitious—but I did read an anecdote in which a group of buddies were riding around town in a convertible, drinking and yukking it up and being generally disruptive. A policeman approached them and told them that their behavior was unacceptable and that they were to settle down and be quiet if they knew what was good for them. The buddies complied and during the entire exchange spoke to the policeman with all of the courtesy they could muster.

The policeman was the only person in the encounter who was not armed.

That's the picture of a law-abiding nation.

Note that the sentencing from criminal offences is likely to affect how even relatively dangerous criminal react to police. If you are already going to face decades in prison for, let's say burglary, shooting a cop is basically nothing on top of that. After all, humans live relatively short time, longer sentences after a while are all basically just the same for the criminal.

So if we compare two situations:
Caught criminal is either likely to be shot by an armed cop or face very long sentence or caught criminal is not going to be shot by cop and will face more manageable sentence, the criminal in first example has high motivation to actually shoot the cop. In the second example, there is not much to win for the criminal by shooting the cop, especially if that crime is still punished relatively strictly.

That in addition factors mentioned in other answers can make the situation feasible.