As MJ pointed out, your brain and body do the work when it comes to Eye Dominance, it isn't something you really need to think about at all or "train specially" because of. The gun will be placed in front go the proper eye because it is dominant. It matters for shotgun shooters using beads because the stock is attached to your face and only ONE eye can be properly aligned with the bore axis. Some people get confused about its relevance to pistol shooting and get too worked up over the whole issue.
In regard to the bigger question of One Eye or Two Eyes open, here is the rule of thumb I give students:
If you CAN use your sights with both eyes open (as in you are focused on the front sight and you can be sure it is super-imposed over your target), you can probably take the shot without your sights and you may be wasting time trying to use them. An example is a chest shot at 7'... people don't need their sights to make that shot, but in a range environment it is very easy to do that both sighted and with both eyes open.
If, OTOH, you try to use your sights with both eyes open and when you focus on the front sight, your target turns into two separated target images and you really can't tell where it is THEN you should close an eye and align your sights properly. So, I recommend that sighted fire be done with one eye only to be most efficient.
Coming full circle, as long as your vision is pretty good in both eyes, it really doesn't matter which eye you use.. with the gun at extension you can use either to get proper sight picture/sight alignment.
In our APH Courses, we experiment with all of this and many people are often surprised at how much of a non-issue the eyes really are in plausible defensive shooting situations.
-RJP