I'm RH and wear it on my right side.
When you look at a lot of the period quivers you start to see some trends. One the plains where the horse was prevalent you see many quivers worn on the left side. Makes sense - that'd be the easiest way to draw while on horseback.
Back East or in the SW you see a whole lot more worn on the right - easier, IMO, to manuever the quiver while walking through the brush. You see the same style in early art from Britain as well worn the same way.
Find what works best for you.