I'd like to throw in the opinion that the moral dilemma "accept ongoing cheating without doing anything" vs. "reporting the student" is a false dichotomy as there are other options besides reporting "the" student.
And I'd like to point out that gut feeling that there is some difficulty in answering the question (dilemma) is exactly right: IMHO it is a dilemma, and options and their consequences need to be considered carefully.
Before diving into the case, let me say: @kingledion nicely commented that honesty and trust are important cultural values - with which I totally agree. But I'd like to point out that this works here both ways: reporting other people (justly) undermines trust. The more so, as you may still be reporting the wrong person, see below. So this undermining of trust happens even if you honestly believe to report the right culprit - plus: all others do not have firsthand knowledge of your honesty.
Even worse, a culture that unquestioningly/unconditionally encourages people to report wrongdoers also allows or even encourages dishonest people to report other people just from spite - including people who did not do wrong. (This is a type of abuse of power; keep in mind that innocent people who are denounced do suffer, even if they happen to be able to clear themselves)
So not only not discouraging cheating, but also reporting a particular student comes at a cost for your society.
I put "the" student in quotation marks because from what I've read in this thread so far, you do not know that that particular student has cheated: Keep in mind that it is very easy to open an account on stackexchange and type in someone else's name to be displayed. And even if a student brags about having cheated, you need to weigh the possibility that they just wanted to sound cool without actually having cheated (false claim).
To be clear: yes, it may be more probable that the student you think of did cheat than that they did not cheat. But you do not have sure knowledge of that, just a suspicion. And if you want that there should be official consequences for the student, solid proof is needed (at least, that is as it should be).
So let me suggest two more options besides "doing nothing" and "reporting the student to the authorities" that tackle the two problems we've identified here:
- Someone is cheating and this should stop
- Homework rules could be improved (reducing not only the probability of cheating, but also that of future students find themselves in the same dilemma you are in, and/or that of a general atmosphere of distrust and denouncements)
You could quietly talk to the student you suspect. I think of asking them whether they know that someone opened an SX account with more or less their name displayed where homework-no-asking questions are actually asked - because according to what you explained here that is actually all you know.
This would give them a fair chance of deciding how to appropriately handle the situation (fair particularly as we've realized now there is a non-negligible risk of wrongly accusing here: a student whose name is misused this way surely has a right to know that this is going on)
My very personal opinion (feel free to differ - but please think about it) here is that the more likely your authorities are to jump to the conclusion that the denounced student is guilty, and the more serious the consequences of a false accusation are and the more difficult it is for someone falsely accused to defend themselves (How do you prove that you do not own a particular stackexchange account?), the more important it is to make sure noone is falsely accused in the first place. Also take into account that even if the accused can comparably easily clear themselves of the suspicion, that until this is publicly clear depending on the possible consequences they will suffer (Make the thought experiment: what would you who never even considered cheating do and feel if you were falsely accused).
As a side benefit, this would give you experience in excercising civil courage. And in practical subsidiarity as in: do not escalate/hand over difficulties to "higher authorities" that you can solve yourself.
OTOH, if the atmosphere is already so distrustful that you don't dare to mention the cheating because you fear you'd be denounced in turn: then I'd agree that talking to the suspect is not the way to go. But then neither should (hopefully!) the instructor listen to any reports about cheating students, and a totally different strategy would be needed to change things.
As I said above, I do see a distinct cost to the denouncement in itself. However, assuming you can do your "duty" against cheating e.g. by directly talking to the student in question, you can think whether your duty against problem 2 may be done without undermining trust as well.
- let the instructor know in their end-of-course evaluation that you deem the homework situation suboptimal due to possible cheating and denouncements by asking someone else/on the internet: e.g. that you personally would have preferred if marks areearned only on proper exams
- if there is no such evaluation, you can talk to the instructor after all marks are given.
If you came to me (faculty) with the "case" as described, I would explain to you that there is no proof against the student (see above), that from your description and a look at stackexchange alone you cannot judge who is dishonest here (the accused student or some [unknown] other student abusing their name) - and much less can I: in addition to the two possibilities you point out, I have to consider the possibility that you are falsely accusing a student you dislike.
I'd then probably ask you what you'd like me to do in this situation.
I'd probably talk to the accused student in private and certainly without letting you know that this is going to happen (see possibility 3) - telling them there's a rumour they cheat and a possibility that someone is abusing their name on the internet (making sure they don't learn your name, neither).
[Side note: in school, a fellow student who was copying from me was once caught. When collecting the exam, the teacher told them (making sure I could hear it): "Next time try yourself, OK?" - which I think was far more efficient wrt. avoiding future cheating than any kind of big fuss and fail mark for both of us.]
I'd think how to discourage this type of cheating in grade-relevant exams.
But then, I grew up in a university culture where grade-relevant marks could not be earned by homework that could be done by someone else (so I'm quite familiar with this type of thought). To the extent that the important exams were oral. Written exams often were "bring all books you like" [if you didn't understand the course contents, you anyways won't be able to extract sufficient amounts of relevant information during the exam].
Homework may earn you a very few points (as an incentive to not completely disregard the homework) but never even half the points to bare passing.
Finally, let me put the "do I have to report some wrongdoing" feeling into some context: over here (Germany), there is no duty to report a crime that has already happened for normal citizens - and there are only rather restricted duties if you learn that a major crime is planned (even in case of a planned murder, you don't have to report that to the police if you report it to the intended victim or [successfully!] prevent it yourself) - though you can report lots of things, and some suspicions you can even report anonymously.
A while ago I read a newspaper article about the local ("county") veterinarian office. They said that probably about 1/3 of the reported cases of cruelty against animals are not concern about animal wellbeing but clear spite against the accused owner. Which causes all kinds of trouble. Not only for the accused and for vet officers who have less time for the real cases, but also because it sows distrust in the neighbourhood. (And another third of the cases was estimated to be honest but unneeded and quite incorrigible concerns [think "those ducklings are in danger of drowning - no you lazy officer, don't try to tell me they are not. I know they are in danger. You need to get going and DO something."])