I'll give my answer based on the science and research. I will also say that I'm a hunter and competitive shooter and my experiences also support what I will offer. Similar to JB, I don't doubt that folk have had success doing it "the other way" (following the handedness, or just closing or blinding the dominant eye). But on the whole the evidence is pretty strong for the following; results are best when you follow eye dominance instead of hand dominance.
When it was studied, shooters in an intramural shooting program were found to do better (score, enjoyment and comfort) following eye dominance instead of following handedness.
When competitive rifle shooters of various abilities were studied, the prevalence of contralateral (shooting off the shoulder that was different than the dominant eye) grew smaller and smaller the higher the level of competitive until at the Olympic level, there were no contralateral shooters.
I have been fortunate to coach hundreds of junior shooters including more than a few collegiate team shooters and a national team shooter, and our experience mirrors the above We for the most part applied the above methodology. The exceptions where we didn't was related to availability of equipment. We had only so many left handed rifles in our program and there were a couple of times where we had too many left eye dominant kids.
I will add a couple of points. It was harder to get boys to switch off the right shoulder to the left. My theory was that they had a decade of running around with toy guns ingrained with their right shoulder (hard to undo a decade of ingraining) The girls generally had no problem shooting off a shoulder that didn't match handedness.
Last point. Closing the off eye (whether it's your dominant or non-dominant eye) is not a good idea. Closing an eye causes a phenomenon called sympathetic dilation where your open eye's iris opens in response to the opposite closure. It seems your brain expects a certain amount of light and will open up the remaining eye's iris to get it. Wider open iris means poorer acuity. Bad if you're shooting iron's. I haven't experimented with glass. But at my age and optical capabilities, I care not to sacrifice any of my visual acuity.
