Only SSN is not public information. They are not published, but the records are public if you go to the right place in person and ask the right questions. (sometimes you have a "need to know" before they will tell you, but it isn't hard to find one)

Even SSN is not private - you need to share it with enough people that it is safe to assume bad people have it.

Connecting an ID to a individual person is very hard. It is even harder when that person is not physically there.