In a Two-Headed Giant game, life gained by your teammate won’t cause the ability to trigger, even though it caused your team’s life total to increase.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time, a “whenever you gain life” ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, and/or planeswalkers at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.

An ability that triggers whenever you gain life “for the first time each turn” won’t trigger if you gain life during a turn before the permanent with that ability is on the battlefield, even if you gain life again later in the turn.

An ability that triggers “whenever you gain life” triggers just once for each life-gaining event, no matter how much life you gain.

If you gain an amount of life “for each” of something, that life is gained as one event and the ability will trigger only once.