Neither Brooklyn nor the Lakers can receive a player in a sign and trade (too much salary on their teams), they can sign and trade their own player though. So the Lakers can sign and trade Dwight, they just can't send him to Brooklyn.
Also recently acquired players cannot be traded in a trade involving other players for something like 90 days (not sure of the timeframe, but it isn't immediate). So Brooklyn could not trade Pierce or Garnett in a trade with other players, but they could trade either one alone (multiple players could come back, but they would have to be the only player out).