Regardless of the T's personal life, this really has no bearing on their ability to counsel and provide good therapy. If you are comfortable and getting the feedback you ask for that is what really counts. Although of course if your T is expressing extreme feelings of sadness or anxiety etc at times then this may affect you. I would in fact mention it if that were the case.
I do more than google if I am researching a provider. You can in fact check them out at the State level and see that they are licensed and other educational things. I do this right away not after I no longer see them. I tell my providers that I research all people on my case. One T actually told me that she researches her clients. Of course usually they have access to our basic medical records anyways.

In theory, we clients are not to think or worry about the T's personal life but to only focus on our relationship with the T and what we are getting out of it. I think it is important to speak up about anything that seems to be out of wack or bothering us in sessions.