I will give an (impractical but I think interesting) answer to when it theoretically makes sense to stall in Bughouse, since on a mathematical level this is a definitive factor in the game.
In particular, let a position on one of the boards be 'winning' when, assuming no further pieces are incoming from the other board, the position is theoretically won with optimal play on that board (given the board-state + pieces in hand for both players). Then there is the following lemma:
Consider players A1 vs B1 on board 1 and A2 vs B2 on board 2. If the position becomes winning for A1 on board 1 (and A2 expects A1 to successfully obtain the win), then A2 can assure the win with no further cost in game-complexity simply by stalling if the clock time remaining for A2 is greater than the clock time available for B1 plus the time A1 will need to force the win against B1.
Why? Because if A2 stalls while having less time than B1, for example, B1 always has the option of themselves stalling, and A2 needs to outlast B1 or else his flag will fall before either B1's flag falls or B1 is mated by A1.
If you (any human pair) were to play theoretically-optimal opponents, expect the game to last only briefly across both boards. Then one board falls silent with stalling; that's when you know that on the other board they've spotted a forced win.