“Psychologists” covers a lot of different people with different sorts of training.
Psychological healing is not done by doing something to a patient that makes them better. They’re trained to give patients opportunities to work on themselves (even when applying drugs).
But to give an appropriately vague answer to a vague question, yes, they’re still human and all humans have issues to work on, and yes the good ones would try to apply what they know to themselves.
But since one of the main things they know to do is to provide listening, I’d think they’d tend to know then to seek that from others, rather than working on themselves only in the company of their own thoughts.