Depends on the game itself and the requirements of the trophies. Some do and don't. However, a definite yes if a save is before a point of time a trophy is about to unlock. Other trophies do pop on load but some others require to do one more time of what is needed to unlock a trophy. For example, to get a trophy that requires a player to do 10,000 kills, you would dl a save that already has that many kills and perform one more kill and voila! trophy unlock if it didn't pop on load that is.