Being a left or right handed archer depends on whether you have right or left eye dominance. It's actually not about holding your bow with your right or left hand etc. The reason why - accurate shooting comes from the bowstring and peep being lined up directly in front of your dominant eye. Often right eye dominance matches actual right-handedness and vis-versa with the lefties. But it's always best to double check what eye dominance you have as it's not uncommon to be left handed but with a right dominant eye.
First, have a look at the photo above. That's what you're about to do - pick small object nearby and look at it with both eyes open. Frame it with that hands together triangle. Now close your left eye. Did the object 'shape shift' or stay in the centre of your hand triangle? Try it couple of times with both eyes alternating closed and open. If the object stays centred in the triangle with your left eye closed, then you are a righty as far as shooting goes. Some people have an even dominance with their eyes - they see an even shape shift with both eyes. If this is you, choose a bow based your right or left handedness and you'll be good to go.