Privatizing Space will do the same thing for it that privatizing the Internet did for all of us. It will take a government-pioneered oddity that was out of reach of the masses and explore millions of ways to make it of financial benefit to the entrepreneurs who can get there first. Like with the Internet, society will be pulled along in the vortex of the Race to Space. Space will become accessible for a price and in ways that would have been inconceivable a decade before.
The technical challenges with space travel are much greater and so the time before we reach the exponential part of the curve is further away, but when the critical mass of Space travel technology has been reached it will be like the Internet in the 1990's. The Internet had been around for 20 years and only governmental and academic types even knew anything about it. Within a few years of its commercialization, its use and uses had exploded. We went from the average joe not knowing anything about it to seeing an abundance of commercials on television for it within just a couple of years.
I mention the Internet, but the pattern is one that you can see in the adoption of transatlantic sailing, rail travel, electricity, automobiles, etc. Space travel will be no different.