I believe that outer space is the new frontier. When the first colonies expanded westward across our great nation, they were moving into uncharted territory. Yes, they were promised land if they could stake their claim to it, but the government didn't 'own' the land, they merely recognized that its citizens were claiming that land for themselves. Those land claims are how the actual property became part of the united states, not by the government 'giving' them the land. If Elon Musk can afford to send people safely to Mars to create a colony, it's the same thing. Claims staked there would be recognized by the nation to which that citizen belongs, be it the US, China, Japan, or Timbuktoo.
I think that one of government's main jobs is to serve all of its citizens. A private company concentrates on it's own profit. Space is a vast resource, and it should be curated to benefit the entire earth in the long term. It is very unlikely that a corporation will chose long-term benefit over short-term profits.
While I fully support space travel and private companies helping to fund said travel, having a random private company be allowed to send someone to space would be a horrible idea. There is so much training that needs to be involved. If a private company decides to cut some corners, or speed up any training for any purpose, it could cost lives. Traveling into space is not something to take lightly. Even the American program NASA has made mistakes and gotten people killed or injured.
I feel like allowing a private company to have exclusive control over something of this magnitude would give way to much power to one person or company. If the government was involved at least there would be some realistic regulations attached to the project. With great power comes great responsibility.