"Mental illness" is a broad term.
I would subdivide:
1. Chronic and almost inevitably debilitating (schizophrenia, brittle bipolar, uncontrolled and destructive substance abuse, severe personality disorder). All hard to treat, and I'd assume that med schools and residencies have interviews at least partly to (try to) avoid accepting residents with these problems.
2. Run of the mill stuff (eg, mild depression, mild substance abuse, mild pd). I'd assume that some of us go into psychiatry in order to self treat (or at least self understand), and I'd assume that one formula for professional success in any academic field would include a little extra narcissism, masochism, and hypomania.
And I think the extent to which PD's are "understanding" can be subdivided into pre match (presumably skeptical, and I wouldn't mention personal mental illness) and post match (when a decent PD would work with you on becoming the best psychiatrist you can be).