How should one decide whether to study logic in a math or philosophy department?
I am a math professor, and from what (very) little I know about philosophy, I would expect the academic cultures to be very different. Once (as an undergraduate) I went to a lecture given by a leading contemporary philosopher. During the Q&amp;A period, one audience member remarked: "I think your point of view is obviously wrong... " and proceeded to elaborate at length! I've never seen this in mathematics. Despite what you've seen in movies, I've found academic mathematics to be quite friendly and relatively nonconfrontational.

My understanding is that (perhaps unfortunately) philosophical logic and mathematical logic are considered entirely separate academic disciplines. I went to graduate school at a university with a strong mathematical logic program, and what my friends were doing was recognizably mathematics. I don't think I heard them talk about philosophy, at least any more than it came up during random conversations. I would recommend to you that you learn something about model theory, recursion theory, axiomatic set theory, the incompleteness theorem, etc. and see if this is something you enjoy.

I doubt very much that you could switch from one department to the other, although I don't have any firsthand knowledge of this.

I would certainly expect graduate admissions and the job market to be very different. My understanding is that the job market in mathematical logic is very difficult; indeed, I only know of one of my logician friends that is now in a permanent academic position. (That said, the others found good non-academic jobs.) However, my guess is that the job market in philosophy is still bleaker.