Having done plenty of searches as a volunteer court-appointed intermediary when I lived in Colorado, I can tell you that state laws vary a great deal. Some states, if you fill out a form with a license plate number and give them a few bucks, you are given the registered owner of the vehicle - and you can then pay a few more dollars and get their complete driving record. Any tickets they received probably will contain their (potentiallly unlisted) phone number. Easy way to get contact info. Then there are states like California, which because of a celebrity stalking/shooting years ago, have tightened up their laws and information is not so easily obtained, unless you have a court-order. And even then, it's a bit difficult. (So if you live in CA, you're pretty safe.)
Knowing all this, I still have shared my Roadster VIN and also my license plate. It's easy for anyone to get anyway. What you should protect, always, is your social security number. That's gold when someone is searching for you. Divorced? Marriage and divorce records are public record. Your divorce paperwork is obtained easily and contains all sorts of information you should be concerned about ... employment, financial status, SSN, parents names, etc.
And of course you should always monitor your credit files, because identity theft is real. I check mine at least monthly and also am sent alerts when there is any new activity on my file.