For contracts to work you need to create the demand and supply contracts yourself.
You can subsidize them, but you shouldn't. Later in the game you will have more than enough civ ships than needed which will slow down the game.
As soon as the civs have at least one freighter and passenger ship, they'll rake in money easily to build more ships. If you have a colony which is set to "Destination" in the Civ/Ind tab of the Pop&Prod(F2) screen (which every colony under 25M inhabitants), passenger ships will fill up the available space. Mostly they don't care how much space is left and just overfill, which creates demand for infrastructure. The freighters will be happy to fulfill it, providing more free space for new colonists. This will go on until the planet reaches 25M inhabitants and can be set to "Stable" or "Source".
Since civilian freighters have less space than most player designed freighters (I go with 25,000 Storage on every freighter module), transports with your own freighters are often more efficient, but more tedious to set up.
So, I normally use freighter modules and colonizing modules, which will sit on my homeworld until something needs to be transported. For the colonizing modules, this means mostly never after the civs establish their first passenger ships.
I say modules because I build them without engines and use tugs, ships with lots of engines and a tractor beam, to move them around. Since they typically spend as much time doing nothing, I'd rather save the resources for their engines.
