Before the client library rewrite we had a special kind of BaseClient called ForeignClient (iirc). I don’t know why it hasn’t been implemented in the rewritten version.
In general, you can just do HTTP requests to their public objects and files (anything else won’t work, as e.g. listing contents of a public directory is not allowed). In order to find the storage base URL, you could use RemoteStorage.Discover, same as the library does when connecting one’s own storage. Unfortunately, it’s not well documented, so getting help on IRC or opening a new thread here on the forums might make sense. You can also just read the source, of course.
Ultimately, I think we need to work towards bundling general sharing code in helper modules, as well as re-introduce the ForeignClient at some point. The latter would be rather easy; maybe you can even contribute it when developing it for your app?