They way the contracts work for now is:
The airlines hire the security company. The airlines pays the security bills. The airlines bills the TSA. The TSA pays the airline. End result: the TSA pays for the security company. The TSA makes the rules now and the TSA decides what goes on planes. The airlines were always the ones with the final decisions, now it is TSA.
This is most likely the way it will run until the TSA can replace the contracted workers with its own TSA employees. This will take a very long time. It is not easy to start a new department of government.