I think it all depends on the attitude, both the teacher's and the student's. If a student is truly out there to find information and to learn more, it's absolutely OK, and the teacher will accept this and even laud the effort. If the student, on the other hand, is constantly trying to disprove his/her teacher and tries to find amunition for diasgreement, then it's not OK.
Likewise, if a teacher knows his/her stuff, s/he should be delighted to discuss whatever the student finds on the web. After all, it's no different than reading a book and asking the teacher questions about it. The teacher will be able to present arguments in favor or against what the student brings up. It is then up to the student to make a decision.
In the end, there are really only two kinds of questions a student can have. First, thera are those that do have a clear and single answer. Any student and teacher will accept the correct answer, hopefully without too much fuss (anything else would be unreasonable). And then there are those issues that don't have a clear answer, i.e. either they are about opinions, or there are several different ways to approach them. In that case, student and teacher should be open-minded and willing to constructively discuss these issues for the benefit of both the teacher and the student and for the benefit of arriving at a satisfactory solution, without involving any ego. That discussion can of course take place in the open.
It may indeed appear to be awkward for a teacher/student pair to lead discussions in the open, with other people participating. Some teachers will feel this undermines their authority. However, authority without cogent arguments is futile, and if one has cogent arguments, one does not need authority. Cogent arguments are authority! (I should have this statement patented

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To be fair, open discussions areprobably much more disconcerting for teachers than for students. Teachers have something to lose, namely their reputation, if it turns out that their teaching is wrong in many instances. So, I would assume that they are much more interested in hiding their identity than students. Then there are of course those teachers who don't care if they are proven wrong, because they are willing to learn and develop. That of course again requires students to accept that their teachers are not infallible and that, ideally, student and teacher should form a "learning team", rather than a unidirectional "teacher -> student" hierarchy.
Wow, that turned into an essay. It's complicated.