Government-owned colony ships and especially freighters are useful. Sometimes you need something transported ASAP (like minerals to a colony with shortage-halted production), and using your own vessels is generally more reliable.
As for encouraging the development of the private sector, the best thing you can do is create a demand by setting up contracts to ferry facilities and infrastructure within the solar system. Once there's the latter in, say, Mars, they might feel a stronger incentive as they can now also start transporting colonists. Also, don't forget to subsidize the companies a bit. At least enough to allow them to buy a ship or two.
EDIT: I did something that may be fixing my original issue. I SM'd a thousand infrastructure on my outermost colony and set up the necessary contracts to have it transported to Earth. A lot of freighters seem interested in the job!