I think you take any part of a medicine job home with you at some points. Psychiatry is no exception. However, as ha often been stated on this board, it's important to have hobbies outside of medicine in general and psychiatry in particular.
After psych residency, after seeing a couple thousand patients, it becomes sort of rote, I would think, leaving you to do what many psychiatrists do - treat the easy stuff easily, and reflect or "analyze" the more complex cases as you need to and see fit. I see many practicing psychiatrists do this.
We have one attending who runs a successful private practice. She takes only patients who pay out of pocket and are "interesting and thought provoking." She does both psychodynamic psychotherapy and medication management.
That is, after all, at least part of the reason we went into the profession.