I think, importantly, that there is no difference. Once the information is out there, it's out there. You can attempt to have a "right to be forgotten" but that's just relying on people to voluntarily comply. If I have a jpeg no force in the world can make me delete it if I want to preserve it.

Public "but only available if you go to city hall" is at least a little more restrictive until Zillow sends out people to start scanning public documents so they can build up their data.

Not public is really the best way to go; and here it sounds like the town has decided that part of the punishment for minor civil infractions is to be put in stocks in the town square, which might once have been enough to shame people into not stealing horses, but now carries a stigma that could last for your entire life.