Earlier quoted context omitted.
It will end up on the feed the court publishes, so it's not like it isn't public information. The police could chill out though and not make a bad situation worse.
There's a difference between the information simply being "public" and a social media account blasting out people's mugshots on Twitter.
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.