I mainly stick to the habit of shooting [handguns] with both eyes open, even though I am unfortunate enough to have a cross dominance issue. Reason being, by closing one eye, you lose half of your peripheral vision, as well as your depth perception...this could prove disastrous when it counts the most, IMHO. To me, making things a bit "easier" on myself [by closing one eye] is not worth the potential cost. Of course, generally speaking, when I shoot I have personal protection in mind. Others may not be in the same boat.
Now, in terms of shotguns, due to having a cross dominance issue, I tend to squint my non-dominant eye, roughly 40-50%, just prior to shooting as, given the long gun platform, keeping both eyes open is just too damn unrealistic in my case (unless I bother trying to shoot left handed). This seems to work for me as it forces my non dominant eye in to dominance..all while maintaining a decent periphery. But, again, I really only use this for doing the shotgun thing. For handguns, again, its both eyes open..all the time.
I would mention rifles, but I have yet to glean much experience with these as of yet. For me, in this case, I guess it would depend on whether or not we are talking about using iron sights or a scope. But, I would imagine going with the "half squint" when using iron sights and keeping both eyes open when using a scope.
In any event, I, like several other posters, am cross-dominant (right handed with a dominant left eye). I find it impossible to shoot with both eyes open, I close my right eye and aim with my left. Works well for me.
In terms of handguns, have you tried simply turning your head towards your strong side
shoulder, thus aligning your dominant left eye with the sights? I use this technique and yes, it took a while to get used to. Now, it is more or less habit.
Some claim this technique works best with an Isosceles stance, but I generally prefer using the "modified" Weaver in most cases. Either way, it seems to work quite well. Of course, you could combine this with a slight tilt of the gun...as long as it is not too extreme.
While I have tried mixing the two, I still generally prefer simply turning my head for proper sight alignment and leaving it at that. The only downside is that your peripheral vision "shifts" to the center/right, given the location of your head at this point. But, you are still able to extend your periphery by quickly looking to the left, if necessary, thus keeping you from having to open/close one eye all of the time.
In any case, there is just something about losing half of my peripheral vision [if closing one eye] that unnerves me.