There are two kinds of private companies.
There's private mining colonies, where as you know you can choose to either tax (and effectively destroy) or buy their mineral output. By default, you tax, which is okay, but in a lot of cases buying is likely better because it's easier to get money than minerals, especially on mining sites with high total accessibility. (The money involved is proportional to the number of mining complexes, not to the amount of minerals produced.)
And there's shipping lines, which over time will come to own lots of transport ships which you don't pay for in any way and which don't use up any fuel. It's important to note that commercial ships will only fly through stabilized jump points. You can use the "Civilian / Flags" tab of the economics window to interact with them.
They've got four types of ship I'm aware of:
Freighters move cargo. If you don't interact with them, they'll move trade goods which are generated and demanded by your populations, especially larger ones. This gives you some tax, and also will give you free infrastructure on populations that need it because infrastructure is one of the trade goods. You can also set up instructions for them to move installations around for you so you don't have to use your own cargo ships. (This costs some money, but makes shipping hardware to colonies painless and free of TN costs to you.)
Colony ships move bulk population. By default any new colony starts out as a destination for them to transport to, though they won't bring people to a place that has no ability to support them (insufficient infrastructure). These always generate tax AFAICT. You can adjust where they'll take people from or to.
Passenger liners are comparatively rare or maybe just emerge later. They seem mostly pointless except for flavor - they move very small numbers of people and make pretty small amounts of money. I'm not aware of anything you can use them for except a minor revenue source.
Fuel extractors mine Sorium from gas giants and generate fuel. Your ships can refuel from them! (This probably costs money.) If they fill up on fuel, the fuel they generate is turned into money and you get taxes.
