When you encounter a skirmish you can choose which side you want to back up and earn that side's rewards. If I earn the reward for choosing one side of a skirmish, can I double-dip and earn the other side's rewards by coming back after the skirmish respawns?
No, you can only earn rewards for one side of the skirmish.
I tested this by finding a skirmish in the Fornis Region near the Great Idalla Ravine. I sided with the Murder of Tirkins and claimed their reward.
I then fast traveled to a different region to make the skirmish respawn. When I returned neither side had a reward.