A shared filesystem is indeed a different thing from a local filesystem, but there's no such thing as a local filesystem repository.
"Local" in the context of "Snapshots can be stored in either local or remote repositories" means "within your network" as opposed to using one of the public clouds. Of course things like Minio have come along since that was written and now the lines are a bit more blurred.
In any case, whatever repository you're using needs to be accessible from all the nodes, and you can check this with the verify API.