Seems a fuzzy distinction. If they require regulation and licensing by the state, then are they purely part of the 'private' domain?
I mean, I don't know about the US, but here, many jobs, whether private or government, that involve dealing with 'vunlerable' groups, require an 'enhanced criminal records check', that would include cautions and very old convictions and, maybe (not sure), even allegations that didn't lead to convictions. Pretty sure that would include things you'd been charged for but not yet faced trial over, and that would affect private employers also.