I think it's appropriate to ask a therapist personal questions if it relates to trust and/or a topic the individual is struggling with, and it would help to know how deeply the therapist understands.

For example, I semi-asked my therapist if she was a narrow minded ideological bigot (only using different terms) by saying that our values don't have to align perfectly, but that I would have a hard time if she were X, X and X. She assured me she was far from that personal profile.

Other than that, I tend to prefer she disclose things herself, trusting that she won't overdo it.