"Linesman worth their salt call fouls. The center referee stipulates what area and when the linesman makes the foul call. But, linesman make foul calls, in fact, the fourth official can also call fouls and does on occassion."
This is a semantics issue, but important for this discussion. Yes, a linesman signals a foul by holding this flag up the in air (and moving side to side to get refs attention, if I can recall my glory days as refing). Then, the center ref calls the foul, etc. If the center ref decides it wasn't a foul, then he can ignore the linesman. The linesman does not have the power to call the foul. He, by himself, cannot blow a whistle and stop the game.
The whole point of linesmen are to assist the center ref in out-of-bounds situations and things he cannot see. All decisions about the calls have to go through the center referee, however.