In Miami I have seen it both ways so I am not sure how they determine which way they dock the ship. At other ports, I would think it all depends on what other ships are in and where the local authorities tell them to dock.
That said, when I was on the Navigator last Sept, we docked the following ways:
Miami - right side facing dock
Ladabee - anchored with bow facing shore
Ocho Rios - right side facing shore
Grand Caymen - anchored with bow facing shore
Couzmel - Left side facing dock