I've come across personal information that a provider might not want in public.  
I tell her and the reaction is always the same.  A bit of a naive shock and then gratitude.  

Just do it at the conclusion of your session.

in the few times i have discovered a providers real info i have passed it on to them and nearly 100% they were grateful.  you can downplay how you found it.  dont tell her it really took you 4 nights in your mom's basement to find it  LOL  

so yes for everyone's peace of mind, let the lady know.

Thankfully, I never did. This isn’t what you asked, but goes to show that some providers can be a little too casual with their identities. I even saw that one of the said providers had left a review at a hotel I was looking into which kind of blew my mind.

Just keep it to yourself.   No good deed goes unpunished.  Seriously guys use your brains.  

If you Googled her provider phone number and her personal Facebook came up, yes, I would let her know. (And I have).

If you somehow stumble across her personal information because you're a perv who spends all their free time searching social media sites for pictures of hot girls, maybe keep that between you and your therapist.

Seriously though, if I found her personal information by researching her provider information, I would let her know so she can take whatever steps to protect her information that she feels is appropriate.

Found a provider's ad. Googled the email in the ad. Second site listed was some data collection site that had tons of her real life info. Talking her name, where she used to live, where she lives now, phone number and even where her relatives lived. With her name, you could find her social media and where she worked.  

I created a brand new email and let her know. I did not want any potential blowback coming my way. She replied and thanked me. Hopefully she is taking precautions