When someone gets arrested and/or incarcerated, is that private information or not private information?

Public information.  Easily searchable.

That is public information.

Police logs are supposed to be public info, arrests are part of those logs, minors are not to be listed. But on occasions, those public records are suddenly hidden. All depends on how much pull one has. This may differ state to state, I'm not sure.

police/court records are public.  some info needs to be redacted such as victim/witness identifying info.  for example, when I had an incident with my stalker, it was witnessed by my neighbor, they redacted my neighbors phone number.

That is in the public domain.