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If you must decline, I would simply say,

"I'm sorry, I don't know you well enough to introduce you."

If you do know them and simply don't want to open up your network...

I'm sorry, but I cant think of anyone in my network who would be able to help you.

Of course, that is the truth, as once you told anyone of your misgivings, they wouldn't be able to help the person.

If you don't want to refuse them outright, tell the person that you will talk to a few of your contacts and see if any of them would be interested in speaking to someone, then call your more friendly contacts and ask.

Don't lie to the person and don't make excuses as that would be both dishonest and unprofessional.

Personally, I do give people access to my contacts, but I am clear with both the contact and the person about it.

Hi, Joe. I met this fellow last night who is interested in your industry, I don't know him, but he'd like to get into widget manufacturing, can I give him your contact info.

Then I get back to the person and let them know what Joe said.

From my own experience below, I've found the safest solution is to, in a way, shift responsibility to the people who the student wants to contact. Instead of flat-out refusing to give them contact details, suggest instead that you will enquire as to how your contacts would feel about meeting the student. By doing this, you are respecting your contacts while not giving your new acquaintance a definite "no". This even gives you the chance to iterate that "many of us don't respond quickly - if at all - to out-of-the-blue requests for introductions". This will also gently lower the student's expectations of a response without coming across as being deliberately unhelpful. Whether or not you actually do reach out to your contacts after this is up to you.

Spend enough time in one profession and you will inevitably gather a network of useful contacts that someone else may want to take advantage of. I work in software development, which in my city, is a relatively small field where everyone is perhaps linked by one or two degrees of separation. With this knowledge in mind, some people have approached me asking if I can put in a "good word" for them or asking if I know anyone in a certain company, which in the case of the latter, half the time I usually do. Most of the time, I believe they mean well and just want to start / advance their career, but I'm not comfortable giving away a person's contact details along with the "Kozaky said I could..." addition. So I just say to them, "I'll ask if they are looking for anyone right now, but you're really better off checking their vacancies page." On a case by case basis, I'll decide if it's truly to my contact's benefit to know this person.

How [do I] politely refuse to put someone in touch with my professional network?

Giving a random stranger direct access to your personal or professional network should feel uncomfortable.

The issue isn't just that your reputation may be on the line by implicitly supporting this stranger, whom you know nothing about; the issue is also that you've not received consent from those in your network to be introduced to this random stranger.

Fortunately there is a tried-and-true method for introductions that sidesteps this issue entirely.

It's called the "Double Opt-in Introduction".

It works by first receiving consent from both parties before making any introduction or sharing any contact information. This verifies that both people are genuinely interested in the introduction and that you're not about to accidentally waste your friend/colleague's time (or worse, help their stalker get their address).

In your case it changes the entire interaction.

Instead of

Random Stranger: do you know someone in [my new field] that you could put me in touch with?

you: Yes, here's some contact info!

or

you: Sorry, I don't know you well enough.

it becomes

Random Stranger: do you know someone in [my new field] that you could put me in touch with?

you: I'll look into it. Do you have a card or contact information so that I can get back to you if they're interested?

With this change in the interaction you're now in control of the situation and haven't had to do something that makes you uncomfortable.

It's then on you if you decide to reach out to others to determine if they're interested in meeting this random stranger.

you to colleage: Hi [name], I met [random stranger] who was interested in meeting someone in [shared field] so I thought you two might be interested in chatting. Let me know if an introduction would be helpful.

If they respond positively, you can then make the introduction. If they respond negatively, you can then send a polite email back to the random stranger.

Hey [random stranger], I reached out to a few of my colleagues working in [field] but unfortunately they weren't interested in a discussion at this time. Sorry I couldn't be of more help, and best of luck.

Even with this final rejection, you're letting the random stranger down easy. You've shown that you've tried to be helpful, and if they want to be angry at anyone they can be angry at the nameless colleagues who weren't interested in meeting them.

In your place I would simply say that I respect privacy of my contacts and I'd suggest the person who's asking to use LinkedIn or another website/event.

The main concern (moral or even legal) is sharing personal information of your industry contacts.

You can ask them what specific aspect of the industry they are interested in, which normally would catch someone who is underprepared.

Then tell them the best way to network is to keep on attending industry events, because they're the best way to really understand the industry and the impact you can make in it.

Finally apologise and say you don't refer people who aren't in your class.

That makes it look helpful, gives advice, and safely answered the question directly.

Otherwise, to use Kozaky's answer as I couldn't put it better myself, for warm contacts: "I'll ask if they are looking for anyone right now, but you're really better off checking their vacancies page."

Good answers already, I'll hit it from an angle.

I'd just refuse unless there was something in it for me. Information and contacts are things you've worked for quite often. No sense giving them away unless you can see an advantage to yourself in it.

I wouldn't be rude, but direct 'Sorry, but I don't do that'.

Up to them how they want to take it. If someone said that to me I'd just reply 'Fair enough' and get on with my life.

(Based on a comment by Thomas, who I think has come up with the perfect solution.) Normally I would agree to just say no, but since your position in teaching means this is likely to come up repeatedly, you can plan ahead. Make a connection with someone who's job it is to filter incoming hopefuls (i.e. a recruiter) and plan that if anyone asks you for a contact, you will send them to this person. They may even have business cards for you to hand out on their behalf.

They are able to weed out the chaff (it is their job) and they will actually be happy that you are widening their exposure.

Refer the enquirer to information that is publicly accessible but not readily discoverable. As an insider, you are in a far better position to "see the wood for the trees" among the glut of information available.

If pressed for contact information, advise your enquirer to consult the relevant public website/profile/directory, explaining politely yet firmly that you are not at liberty to give out private contact details (no reasonable person would be offended by this; if the enquirer is, he/she really needs to learn the rudiments of GDPR and/or other relevant data-protection legislation, as a matter of urgency).

In general, a fellow professional will not appreciate your giving out a personal telephone number to a stranger, unless the stranger were a potential customer... personally, as a freelancer in the UK, I am happy to receive a call from a stranger interested in engaging my services (and prepared to pay my rates -- in general, I do not tolerate attempts at haggling); on the other hand, I am definitely not happy to receive a call from a stranger trying to sell me something.