Are psychiatrists allowed to see other psychiatrists or have mental issues of their own?

Yes: Many people have concerns and sometimes serious illnesses including psychiatrists. I am so happy they receive help and some come to me and i respect them so much.

Why not?: I have known (and also treated) many excellent psychiatrists who also have issues like depression, anxiety, and post-traumatic stress. In general, these issues do not adversely affect patient care. When severe though, the psychiatrist may take time out from work. If issues are untreated, or there's a relapse with substance abuse, etc, the situation is very different and requires intervention.