An ability that triggers "whenever you surveil" triggers after you're done surveilling, even if you have fewer cards in your library than the number of cards you're instructed to surveil. It even triggers if you have no cards in your library.

If you surveil before Whispering Snitch enters the battlefield, surveilling again during that turn won't cause its ability to trigger.

In a Two-Headed Giant game, Whispering Snitch's ability causes the opposing team to lose 2 life and you gain 1 life.