Keep Your Mouth Shut
The absolute best advice for keeping private information private is to keep your mouth closed. There is an old saying that is an excellent reminder of good privacy practices: Even fish wouldn’t get into trouble if they kept their mouths closed.
In almost all cases, private information is first shared by you. The best way to keep information from spreading is to stop it at the source — by not sharing it with anyone. Unless someone has a specific need to know, they don’t need to know.
This makes you, ultimately, in control of your own privacy.