How do professors talk to students, how do students talk to professors? And what about written communication like e-mail messages?
When I was a research assistant and taught programming courses, I used to talk to students on a first-name basis and they also talked to me in a rather informal way, both face-to-face and in e-mail messages. I did not ask for permission, though. At that time, I was only five years older than them and some students were even older than I.
But time changes, and now my students start to be only half my age. As a PostDoc in Basel I still talked to students in an informal way addressing the group as "Ihr" and not as "Sie" and addressing single students as "Du." At the beginning of the course I asked students if this would be OK and they agreed. I did not explicitly offer that they could use my first name, but I was fine with that. Of course, in face-to-face communication, you can often avoid to directly address the other person. Although most of the students talked to me rather informally, the majority did not use my first name in written communication. Here they were back to "Frau Mahlow" and "Sie." So I got messages starting with "Hallo Frau Mahlow" or "Liebe Frau Mahlow." When answering them, I used the form they had chosen in the first place: if someone addressed me as "Frau Mahlow", I answered in the formal way, but would still talk to them using "Du" instead of "Sie." Sounds complicated? Yes it was.
When I started substituting in Konstanz, I decided to completely switch to the formal style. So I would communicate with all students in all situations addressing them as "Sie." That's rather simple. However, in Germany, academic titles seem to be more important than in Switzerland. So I get messages starting with "Sehr geehrte Frau Professor Mahlow" (although I am an acting professor, I'm not allowed to use this title), "Sehr geehrte Frau Dr. Mahlow", "Hallo Professor Mahlow", "Hallo Dr. Cerstin Mahlow" and so on. Some salutations read rather odd and you can feel that the student was unsure about how to address me -- the English speaking student addresses me as "Dear Dr. Cerstin."
When I started my academic career in Zurich, I thought it was rather awkward that my supervisor was on a formal basis with all students, but as soon as they graduated he offered to use the informal communication form. Sometimes as last thing in the oral exam. But now I think that this in fact a good solution; I changed it a bit and with my student assistants I also communicate on a first-name basis before they graduate. It feels comfortable this way.