Generally, no, they have a license, generally a broad, perpetual, non-exclusive license per the conditions attached to use of the site. not ownership. OTOH:
> I thought the core of section 230 was about exactly that - that a site operator does not constitute a "publisher" for everything that is written on their site, if the site is just a "dumb" software that displays content written by the site's users - and therefore can't be made liable for anything written in user-generated content in the way they would be for editoral content.
This is wrong on multiple levels; it specifically does not require that it be ”dumb” (neutral) software, which would have made them a distributor not a publisher under the law without Section 230; it specifically is to allow site operators (and other users) to do things that shape which content is presented without being subject to liability as a publisher.
But that's about liability, not ownership.
> (... this doesn't even touch on the situation where that content contains intellectual property or PII.)
If its not IP, no one has ownership ; ownership of content only applies to IP of some kind.