Zach Braff had a really great monologue on SNL last night (Video) with an awesome song about New Jersey. Off hand he mentioned the Statue of Liberty and how it's really in NJ. This is something thrown around a lot in NJ and it renewed my interest about topic.
After pouring over the Wikipedia talk page for the Statue and sorting through arguments over the original 1664 charter and interstate compacts from 1834 I think I have the wording that makes the most sense.
Liberty Island is physically located with in the borders of Hudson County, NJ. But the island itself is federal property, belonging to no state, but federally managed through the State of New York.
So really no one owns it. But it is in NJ and it can be considered part of NY since the State over sees the running of the island. So in most ways everyone is correct.