Members of any given instance can access (read, reply, boost, favorite) posts on other connected instances as if they were on the same service; this is called "federation". Mastodon uses the GNU Social protocol, and can therefore federate with any site that supports that protocol even if it is not running Mastodon.
Usership of any given instance -- i.e. who is permitted to be on that instance -- is directly controlled by the admins of each instance; there is no centralized authority. This structure helps greatly towards containing and limiting abuse and trolling.