In general, if there's a record of something which was captured in the course of ordinary business which is relevant as evidence in a court matter (such as the recording of your Ring camera), and parties to proceedings have good reason to believe you have this record, then they can generally get a subpoena issued to compel you to produce it for the court. This applies to both the prosecution and the defense (both criminal and civil).
The protection against "unreasonable" search and seizure comes in the form of the fact the requesting party has to convince the court (usually the registry) that there is reasonable grounds before they will issue a subpoena.
As an investigative matter (prior to any charges, court listings, and subpoenas), it is possible to get a search warrant including for evidence held by 3rd parties who aren't suspected of anything. Again, police don't have carte blanche. They need to convince a judicial officer of some sort that there is reasonable grounds before a warrant will be issued.
There are ways to challenge a warrant/subpoena. Sometimes a successful challenge only serves to make the evidence inadmissible but doesn't prevent the search in the first place (aka "you can beat the ticket but you can't beat the ride).
All that said, some judges / courts tend to practically be a rubber stamp for whatever warrant / subpoena the police want. Others actually do their job. It ain't perfect, but if you can think of a better system, I'd love to hear it.