I know that everyone is different, and I'm by no means an expert, but I can assure you that in my case, it does make a difference which eye is dominant EVEN when shooting instinctively. I think that you should ALWAYS start out shooting from your dominant side. Sometimes, your brain may not be shooting instinctively, even though your eyes "think" you are.
I have been a RH left eye dominant shooter for over 25 years but just picked up traditional archery around 2 years ago, 2010 was my first hunting season. I started out RH as I always have and am in the process of making the switch to LH right now after missing a very nice 130" plus whitetail in Oct, from the ground at less than 15 yards!!
I forgot to mention that I missed him by nearly 3 feet as my left eye took over and directed the shot even though everything "felt" just fine. I keep both eyes open at all times like you should.
I don't understand exactly what happens with the interaction between the brain/eye/body but I can shoot targets accurately from either side even though I'm understandably still a little better from the RH side due to my 25 year head start :D The problem for me comes during hunting situations. When my "instincts" take over and I'm focusing on making a clean, ethical shot on a live animal, no matter how hard I try, I can't convince my brain that my left eye shouldn't lead the way, after all, it is dominant, has better vision and doesn't have a string in it's way either!! It's very hard to overcome your body's natural programming and I'm starting to believe that if shooting with both eyes open, it can only be done consistently in a target shooting environment when you're relaxed and have time to work out in your mind how to compensate for your cross eye dominance. Shooting from the wrong side has the potential to have a negative effect on every cross eye dominant shooter at some point. I know guys that seem to have successfully worked around the problem by using a patch or closing an eye to take one element out of the equation. That works ok but adds a different set of vision problems.
I almost gave up on traditional archery until I decided to make the switch to LH shooting last year. The switch isn't easy and I'm not all the way there yet but I'm working on it. I still shoot targets from the RH side but I will never take a RH bow in a deer hunting situation again. I have learned my lesson and I believe it's unethical for me to take a weapon into the field that I don't consider myself to be completely competent with at all times. Traditional archery is difficult enough that I think you should always do what you can to stack the deck in your favor and eliminate all potential problems before they arise. Sorry, I forgot to mention, I think you should switch your son to left handed shooting :D
Sorry for writing a book, but this has been a struggle for me and I know that I can't be alone in my feelings on this.
Take care,
Jasper2