If you receive your paycheck electronically, the transfer likely occurs over the ACH system, the primary operator of which is the Federal Reserve. The Fed also runs FedWire. The Federal Reserve has actually been at the forefront of electronic payment and settlement systems since their advent.

Though, the Federal Reserve is quasi-public/quasi-private. But for any systemically important transactional system you can be sure that the government is plugged in one way or another. The Clearing House company is the main alternative to the Federal Reserve electronic systems for interbank transfers (they're the "private" ACH operator), and they're heavily regulated, with the Federal Reserve given substantial oversight authorities.