It's complicated and varies depending on the broadcasters.
Let's take a Stanley Cup Final game. You'll have three major outlets (American, Canadian English, Canadian French). Each will have its own mobile unit (or units) at the arena. (Rare these days is the network that actually owns the trucks, by the way. They're nearly all leased.)
There are only so many camera positions in an arena. The lead play-by-play camera (usually called the "follow" camera because it follows the puck as a wide shot from center ice) will be shared by all. Who actually runs the camera often depends on where the game is played. But it's shared. Some other cameras are also shared for replay purposes, but the follow and the others are all fed into the separate trucks and all are available to the individual directors at all times. The NHL supplies the over-the-net cameras for replays of pucks on the goal line.
Each outlet also has a number of cameras to itself. These unilateral cameras are also used for replays, bench closeups, and so on. The audio is also complicated, usually matched to the camera on air at the time.
There are also international (non-North America for NHL) feeds. Those broadcasts have fewer cameras of their own and take a feed from a major outlet for the most part.
Once upon a time, when local telecasts of the final were allowed through the early 90s, and there were two Canadian rightsholders in each language, it was even more involved. The Canadian rightsholders essentially alternated, but a Final Game 7 would mean CBC, SRC, CTV/Global, TVA, ESPN and potentially two U.S. local outlets would all be in the arena. Seven feeds of one game. You can see why cameras were shared. (There were five or six originations of the 1971 Stanley Cup Final Game 7 in Chicago: CBS, CBC, SRC, Hughes, which may have been originated by WSBK Boston, and the Blackhawks theater TV feed produced by WGN).
A regular-season or early-round playoff broadcast may only have a couple of trucks (home and road), but the follow camera and a few others is still shared.
Prior to the rise of cable telecasts, a road team would be assigned a single camera for cutaway shots if the home team was televising (usually Canada, New York or Boston). There would be a single truck, and an associate director, while listening to the home team director and road team announcer (and in contact with the road team's station control room on a separate phone line), would cut to the road camera going to commercial or when the home team would put up graphics for the next game, etc. In those days, intermissions were handled back at the studio. The road camera would shoot the scoreboard clock so the station would know when to cut back to the arena.