I guess it depends on their client load and how much are they curious. Perhaps if you're just an ordinary depressed / anxious client they won't do it. But if they think there's something unusual /special about you, they might do a quick check.
I'm a private tutor and I usually do a very quick fb search for my students or their parents. Nothing deep, just a one minute check of their public info. The reason is to get a general picture of the student's family environment and interests. I wouldn't try to look up their /their parent's forum posts or anything like that.
I guess it's similar for therapists.