is star spring once per turn per player or once per turn for both players?

Per player, not once for both players. Star Spring triggers on "the first time a player plays a non-token unit here each turn" — that condition is checked separately for each player, so your opponent playing a unit there doesn't use up your trigger for the turn (FAQ #12439).

Whoever plays the unit controls the trigger and decides whether to move one of their other units at Star Spring back to base. Note that only an actual play counts — moving a unit in does not consume a player's first-time trigger.