There are international treaties already in existence that every space capable nation has signed onto, which prevent exactly that from happening.
When Musk or Branson or Bezos puts a rocket into orbit, they have necessarily gone through an elaborate process of securing approval from the U.S. government. That approval won't be granted unless there is a detailed, step by step plan for everything that will happen on the flight. That not only includes the exact flight path, but also what personnel and cargo will be on board. If there is anything that the government doesn't like, for whatever reason, that permit is denied.
If Musk decided he wanted to put a small satellite into orbit. that would need special permission. If he wanted to take some rich tourist on a joyride, that would need special permission. If he wanted to send a robot probe to a piece of space rock in the asteroid belt and claim it as his property, it would automatically be denied, because that is against international treaties signed by the U.S.
Rich people claiming bits of space for themselves as their own profit won't happen because the government of every space capable nation on Earth will always be motivated to retain that control for themselves, out of "public interest".
