I really wouldn't like the stuff that you're hearing about binge eating and not having many friends if it's current stuff. Wouldn't mind at all if it is from the past.
Last week a woman died of the eating disorder that I used to have and that I now am involved with professionally, my colleagues knew her but I didn't. She was the same age as me but left behind a tiny baby

this stirred up some pretty awful feelings in me (guilt and confusion at why I've managed to be in recovery for many years now) which I could only say in my therapy, because of what I know about my colleagues and their struggles and how it's all too recent and too raw. I know that in the past, my therapist has had her own demons with disordered eating, but I was fine bringing it up with her because I believe those days are well behind her.
I feel as though I know a substantial amount about my therapist's mental health, and to me it is sensitive and personal information. For the most part it's been very useful to know that somebody else could've been in a similar place to me and escaped.
What I will say is that in my opinion, knowing a lot (or what feels like a lot) in general about your therapist makes it harder to reconcile the idea that it is purely a professional relationship with no place outside the therapy room.