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This would fail on many levels.

In order for this to work for any time at all, you'd need to deal with any cat population.

Upon doing that, the population of ALL rodentia would explode. They would then attack the food base, and rats are not picky about what they eat. Your trained ones might be able to be handled, but all the others wouldn't.

Then, there's the whole privy matter....

These are a few diseases from poor sanitary conditions, and rodent borne illnesses.

Dysentery.
E-Coli
Plague
Hepatitis

FULL LIST HERE

And every other disease related to poor sanitation.

This would be a good way to introduce a civilization that crippled or killed itself, but not a good system of messaging.

So basically this is a "star" network where the data packets are rats. https://en.wikipedia.org/wiki/Star_network

The Exchange is where the interesting problems are, and it also is your bottleneck.

Musing about LOGISTICS.

THE TUNNEL NETWORK

If we're talking about a city of 100,000 rats -- er, households, then you might have a link-and-branch system of ever-widening tunnels until the major conduits empty into the Exchange. I suspect that you would need a reasonably geologically stable city.

THE OPERATORS

You'll need many operators, related to the number of messages per hour. I'd wild-guess that 1 operator could reliably handle 15 messages per hour for eight hours. If each house is expected to send one rat per day, then that's 100,000 rats in 24 hours -- round that up to 4,000 rats per hour. So you'll need 270 operators working every day... maybe 150 of them during the waking hours, and the remainder divided between two night shifts. Or whatever!

TRANSMISSION ERRORS

For various reasons -- for example, bad handwriting, or failure to pay one's bills -- the Operator may return rats to the sender with a "not delivered" tag. How sophisticated this feedback system becomes depends on the efficiency of the system and the needs of the population. In particular: the more commerce that flows through the network, the more sophisticated the messaging system will become.

THE OFFICE

Note that if your peak work hours requires 150 workers, then assume you need perhaps 15 supervisors (?), a dozen or so troubleshooters (now there's an interesting plot setup!), and a triumvirate of Bosses to solve really big problems. Plus an above-ground Customer Support Centre with a row of customer service experts whose inter-personal skills rival your Department of Motor Vehicles. And their Supervisor. And a Director (who is too important to have an underground office). So this Central Exchange has airspace and workspace for maybe 200 people per 100,000 population. Maybe. Scale to taste.

THE RAT STACK

You'll need more than one trained rat per household: for example, the house will need one on standby for sending a message, but also the Exchange will want AT LEAST ONE on hand for dispatch. And this is where your city will scale its rates. If you order more than one Central Dispatch Rat due to the heavy volume of orders to your rat-packet-home-shopping-catalog, you'll have a heftier utility bill.

But anyway.

Say the default is one Central Rat. That's 100,000 rats in 100,000 cages in the Exchange, probably sorted into some sort of rodential Dewey Decimal system. Which means the workers in the Exchange are skilled workers; not only do they have to be literate, but they also have to understand the System, and be able to FILE and SEARCH the system with a very low number of errors. Imagine the pain of a mis-filed Rat that gets dispatched, ostensibly to your home, but instead ends up at the Prime Minister's estate. We are not amused!

RAT HOLDING

As an aside, the Exchange will not want to immediately return any rat to its home: if the Exchange's cage for that home is currently empty, they might want to hang on to it. This is where errors will happen, because someone's mis-filed Rat will cause another household's Rat to be held at the Exchange.

PROBLEM MITIGATION

Thus you will need a Rat Mitigation Strategy. With a skilled workforce and reasonably efficient checks, this will be minimized, and therefore just a cost of an efficient network. In careless cities that have cut back funding, hilarity may ensue.

Toss in at least some paranoia or threats about a Virus being released into the network, and there will occasionally be political and popular backlash, but if the network is really invaluable, such problems will be paper tigers. In the paper RPG culture, this is what we call a "plot hook".
Assume that occasionally a householder will end up in therapy due to Rodent Anxiety. This is what I call "local color".

SECURITY

Finally we have the fact that such a network, if efficient and effective, will be quite valuable to a city... and therefore it is a single point of failure that needs to protect itself against hostiles. So expect to have security forces -- or just security theatre -- posted there permanently. Perhaps you could place the Exchange close to a military barracks, for a "free" level of security.

You're limited by the rat's attention span. It isn't that they aren't trainable (in fact their memories can be very long indeed), it's that they're easily distracted. While they might memorize the route, they might get distracted by other things along the way and forget about their task.

The other thing about training is that you might need people in/near tubes to train the rats.

One central place might not be best. Instead, there might be a series of central places covering a particular area, and then each gets handed to the next place and so on. I think it's going to be a lot more complicated than just one place, because the routes have to be short.

Moreover, you are going to have to build hundreds of miles of specialized tunnels for your rats, a huge expense in infrastructure. There has to be room enough for the rats to pass each other, but not so much that they might get distracted.

Other logistical problems, if there are specialized tubes for the rats--clean up. Rats will pee and poop as they go, so someone will have to clean that up. They are also going to end up being more of a vector of disease because people handle them more.

Possible point of failures:

• predators: rats running up and down will attract attention of predators, possibly terminating the transmission. Example of predators: weasels, cats, owls.


• pheromones: a single female on the season might distract ordes of males, rerouting the transmission to unwanted directions.


• latency: in cases where a certain address is highly involved in communication, it can ran out of carrying rats (either to send or receive). This would end up in a denial of service type of failure.

As Mentioned to Problems with this are many-fold. I will try not to be too redundant.

The main problem I see with this is that the cost and work required to train/maintain/staff and problem-solve this are huge.

As someone already mentioned the cost of hiring skilled clerks/trainers to maintain all this seem way too inconvenient. Bottom-line I just don't see this being more effective then its main competition. Human runners, which have been used for millennia for their speed, and ease of use. No real education required. A noble house having their own personal messenger, that is also just a serf/slave is way more convenient and cost effective, then this complicated "Rat System"

Well, for one roman plumbing doesn't quite work that way. all the waste didn't go "back to central" it just went out into the country towns which used it for fertilizer. So I suppose you could use rats in the clean water pipes. Though that isn't a lot better than
rats running in black water. Maybe rats could run on the streets? After all, Romans are known for roads as well. The rats could have a central dispatch per neighborhood. And a larger regional dispatch that uses people to go from neighborhood to neighborhood.
After all rats are relatively intelligent and would probably be able to be moderately reliable at delivering messages if they are cared for properly and like their owners.

It's probably not worth the hassle of even training them because a rat's lifespan is about 12-24 months

This has all the pros and cons of homing pigeons

And therefore, it's believable. Homing pigeons were occasionally shot by hunters, killed by eagles, didn't end up where they needed to go... but the system worked for what it was intended for hundreds of years.

Pros

• Cheap and easily replaced


• Highly trainable


• Reasonably hard to stop once in flight

Cons

• They have the brains of pigeons


• They need to be fed, and you can't let them do that on their own or they go feral


• They don't produce a high-quality fertilizer

Having demonstrated that the idea works in concept, what remains is to use the concept's strengths to move your plot along and it's weaknesses to inject humor into your story.