If Nightveil Specter dealt combat damage in a two-headed giant game, do I choose which player's top card to exile or do I exile both?
You only damage one player, so only one player will exile a card. You get to chose which player you damage in the Combat Damage Step.
810.7a Each team’s creatures attack the other team as a group [...]
810.7f As the combat damage step begins, the active team announces how each attacking creature will assign its combat damage. If an attacking creature would assign combat damage to the defending team, the active team chooses only one of the defending players for that creature to assign its combat damage to. Then the defending team announces how each blocking creature will assign its combat damage. See rule 510.1.