The island was purchased in 1864 with gold from the Hawaiian king by a Scottish family. The king saw the writing on the walls then and asked that if/when Hawaiians are at risk of falling to imperialism, to let them have it as a backup. And here we are, 150 years later, and the Robinson family has more or less stayed true to their word. About 100 natives live there with limited technology, no police, and I believe no money. Since the polio epidemic in the 1950s, outsiders have not been allowed in. Its Hawaii's 7th island, often ignored because tourism is mainly limited to boat rides near it. Only the Robinsons are allowed to bring guests. Here are some bits from this website:
On Ni‘ihau, islanders speak their traditional dialect of Hawaiian, and the entire island operates in a completely self-sustainable way by growing, foraging, and hunting for their own food.
Land was not considered property to natives of Hawaii, and this allowed the islanders to live freely, without having the responsibility of renting or leasing land. Wild sheep, boar, and cattle roamed the land as well and were merely considered neighboring inhabitants of the island.
From my research, there are various ways they make money from the outside world, mainly selling seashell leis (necklaces), and some limited tourism like helicopter visits and hunting safaris (like in Africa). With that money, they have some outside technology like houses, solar panels, and bicycles.
Is this anarchism or monarchism? The Robinson family still owns the land and could cash out if they wanted any time. I have no idea what they do now. Its two older brothers, who might even live on the island amongst the natives. Anyone know more about this arrangement?
The article oversells how closed off the island population is. They regularly travel to Kauai, and there are visits from US Navy personnel too. But, it is functionally closed to haole tourists as well as most non related Hawaiian residents.
The article also oversells the roles of the Robinsons as benevolent overlords rather than focusing on the actual inhabitants. The residents regularly buy and sell things on Kauai, selling crafts, etc.
Now, as to your question on the actual arrangement between the Robinsons and the natives, that's past my scope of knowledge.
Also, worth noting that the island has no water except for rain, so it's not likely it could ever hold a huge population anyways.