I am sorry your coworkers are jerks. They should absolutely not give your information to anyone. Especially about what school your kids go to for Pete's sake.
We had a manager on one of the units where I work do a "Doc of the week" board. I saw info on there that was way over bounds. Like if they were married, had kids, where they were from, etc. A supervisor saw it one night too and we both said that gave away way too much info. It hasn't been done since. I was told the docs were asked what they could put on there, but I still thought it was over the line. Especially mentioning kids and thier ages and their spouses. I am so glad that was stopped. I live in an area in MI that is mixed Trump country. While our county didn't vote for him, a lot that live here did.
Katie I am so sorry that in your area you are experiencing this. Management needs to put a stop to this immediately.