I understand your thinking and if we where talking about a kid or someone who did not have a business that put this info out there on their own then yes I would agree with you. However in the day of the internet most trainers have web sites and advertise in that manner. The info and pictures are out there. I have given my trainers info out to quite a few people over the years and he has thanked me for that several times. It gets him new clients. Ones he might not have if I had not given out that info and 99% of it was done on the net.
She mentioned the trainer's first and last names, AND the name of their new son. That to me (as a parent) sends out all sorts of red flags. While I still believe in the privacy of my instructor, sharing her child/children's names or other info is completely inappropriate.