Excellent answer and of much help, QtM (or perhaps that should be Dr. QtM :D). Thank you very much for going to the trouble of asking, and please thank Dr. Carl for me if you get a chance.
His response puts me at ease, because my own thinking was along the lines of his wonderfully detailed answer: that it would indeed depend on both the form of treatment/therapy, but also the specific doctor/patient relationship.
It seems to me that with some patients, the pdoc might offer the use of his/her first name depending on whether the pdoc suspected this particular patient might be more comfortable by doing so (if, say, the patient had a fear of authority figures or doctors specifically and seemed to be ill at ease). I think a sensitive psychiatrist might say something like, “please feel free to call me Jane if it makes you comfortable” in such circumstances.
I don’t think it would necessarily be appropriate for a psychiatrist to tell the patient to use a first name, but offering it as a possibility – that’s left up to the patient – seems like a kind thing to do. At least if the doctor senses that this might be better at making the patient more at ease. Sort of like what picunurse says: in some cases, the doc might want the patient to feel this is a collaborative effort. At other times this might not be effective.
Anyway, I’m glad my instincts were correct, at least as far as it not being a cut-and-dried situation.
Personally, I’ve had a few different talk therapists, including one who was a psychiatrist, and a few meds-only psychiatrists. All of the docs were Dr. ____, no matter what the relationship was. But with the talk therapist pdoc, whom I liked very much, I felt a bit awkward referring to her as Dr. ____. I never had the nerve to ask her if I could refer to her with a first name, however, because I thought she’d be insulted.
(But this question doesn’t actually relate to me – it cropped up when I was writing a scene with a fictional shrink and I became curious.)
Anyway, thanks very much for all the answers so far!