The "accepted" test for eye dominance requires a couple of steps.
The most common instruction is that you hold up your thumb at arms length and with both eyes open, line the thumb up with a distant object. Then close one eye.
After you close one eye, if the thumb is still lined up you closed the non-dominant eye. If the thumb/object alignment "jumps" to a different alignment you have closed the dominant eye.
This test is almost universally taught to all precision handgun shooters, since aligning with "the wrong eye" will cause "flyers" that may miss the target by a very large distance.
Beginning shooters are often taught to "block the non-dominant eye," with an eye patch or some other device to avoid alignment accidents, but once using the "correct" eye consistently is learned, blocking of the other eye is usually discarded, although a very few experienced shooters may continue to "patch" if their dominance is very weak.
Most experienced handgunners shoot with both eyes open, but consistently "form the sight picture" from the dominant image.
The degree of dominance is variable, and I've known a couple of people who claimed that one eye was "usually dominant" but that the other eye sometimes "took over" under different conditions. (Usually fatigue or eye-strain related?)
A few people have claimed that one eye is dominant for distant viewing, but the other takes over for near objects. In the one or two such persons from whom I've heard this claim, the problem went away after they got new (properly prescribed?) corrective lenses in their glasses.
Good handgun shooters with cross-dominant vision are fairly rare, but probably in about the same percentages as for the general population.
I've known a couple of "experts" who fired right-handed using their left eye with no noticeable difficulty. I've seen no (competent) reports of any accepted theories that cross-dominance is associated with any maladjustment or other effects of personality.
The somewhat faddish popularity of "3-d" pictures and posters, in which you attempt to "hyperfocus" on scrambled images to "see" the two separate images as a "sterieo picture" has been implicated as a cause of deterioration in vision, "damaging" eye strain, and in a few cases some (probably temporary?) psychoses. There was an appearance of "popular art" of this kind ca. early 1950s, and I've seen a couple of advanced text books that attempted to use the "feature" to print three-dimensional graphs of "electron trajectories" and the like, but the practice didn't last long and appears to have faded due to complaints from ophthalmologist and psychiatrist associations who made credible claims that it was being observed to cause harm.
There have been a couple of cycles of reappearances of the 3-d posters/pictures. It appears that only people with "weak eye-dominance" see the 3-d images easily, hence easier visibility for people with "young eyes" (not necessarily related to the persons' ages). During the earlier appearance, "wedge lens" glasses could sometimes be found to assist others, but I've seen no such aids during subsequent cycles of the fad.