To give you a relatively uncomplicated answer:
Radio networks and to a lesser extent stations record these games and cut highlights (or in certain cases get highlights directly from the flagship stations afterwards).
Typically, production assistants who work in the tape room will call up the flagship station and ask for a coupler, which is nothing more than a phone feed of the game (like if you call up a radio station and they put you on hold you get to hear whatever is on the air). In the old days, FSR, ESPN or your local station would have to use reel-to-reel machines to roll on the games; nowadays, with the advent of digital technology networks and stations are able to roll on more games and cut highlights in a more efficient manner.
And yes, there are many stations that will not allow their competitors to roll on their games. But for the most part, stations will help each other out as long as the highlights are properly credited.
I hope this helps. Any other questions, please feel free to ask.