The first thing we do in NASP with 2.3 million kids this year is to determine eye dominance. We then strongly recommend but don't insist, they use a bow that matches their draw hand to the dominant eye.
This allows the archer to keep both eyes completely open for better depth perception, peripheral vision, and balance. If a person shoots "wrong-eyed" they often have to fuzz up or close the dominant eye to avoid windage misses.
I've witnessed many teachers and veteran archers switch from RH to LH during our trainings who do much better in just a couple of hours. Not everyone will put their heart in this dramatic change though. In archery, both hands do something important, and for the new archer, completely foreign to both hands. Matching the eye and string is more important than which hand holds the bow.
I'm RH and left-eye dominant. I shot RH for 28 years. I've shot LH since 1996. I switched not because I was having trouble with accuracy due to the eye/hand issue, I had extreme target panic. I tried a LH bow and the panic was and remains gone (thanks to God because I thought archery was soon to be lost for me.)
When I shot RH I had to shut the left-eye. I thought (maybe?) I was an instinctive shooter in the late 60's through about 1983. I think I was a gap shooter though. I didn't have any trouble being accurate. I used to shoot pennies off my sister's troll dolls at 33 yards with a RH Bear B Mag (the length of our sub-division backyard).
I check my grandchildren's eye dominance as soon as they put down the bottle. I get the bow that fits their eye dominance. They won't know any difference. (I need someone to hand my LH recruves down to!)
By the way, I'm pretty strong Left-eye dominant. However, during some shooting sessions that dad-burned right eye wants to chip in too much and I miss. Some times I have to wink the off-eye, let the dominant eye do its job and then open the right eye. I suspect sometimes, with certain bows the riser blocks a bit of my left-eye vision and the right eye takes over.