Eyes, as in plural. Using your natural binocular vision reduces eyestrain and helps you shoot better.
Different sight systems demand that the eyes do different things, but whatever system you use, shooting with both eyes open will help you hit the mark.
Thanks to the fact that we have two eyes that face forward, we are blessed with binocular vision. This gives us great field of view and also enables us to gauge distance. Doubt that? With a patch over one eye, try walking through a forest or up a rockslide or even across a banquet room full of chairs pulled at random from the tables. Try estimating the range to a deer in new terrain or a hawk in a distant tree.
While people who've lost an eye can adapt remarkably well, there's no question that two eyes give you an edge. Your eyes are engineered to work in concert. One assesses the size of an object and brings it into sharp focus as the other brackets intervening space. The operation is complex and subconscious. But the result is much preferable to monocular vision.
With double the effective pupil area, you also double the light your brain receives to construct an image. So it is perceived as brighter, or better lit. When you squint to block excessive light or give your dominant eye full rein, you induce eyestrain because one pupil is trying to constrict (to block light) while the other is dilating (to admit light denied it by your closed lid). Squinting tires your facial muscles too.
Odds are, you squint when you aim through a rifle scope, but as in shotgun shooting, you're better served using both eyes to direct the shot. Squinting while aiming is most natural but not a good habit.
Because high-power scopes and very restrictive target apertures all but demand a squint, some target shooters have worn patches over the subordinate eye. The patch cures the squint and thus relieves the strain on facial muscles. The eye strain remains, however, as the covered pupil dilates. A better option is a piece of translucent Scotch tape over the shooting glass lens on the subordinate eye. It will blur the target while still admitting light.
With a low-power scope, eyes-open shooting should be easy. You see the reticle with one eye, the target with both. There's nothing to line up; reticle and target appear in the same place. Learning to shoot this way is just a matter of practice. After a few successes, your eyes go to work automatically. That's good news; when hunting you don't want your subordinate eye handicapped by a strip of translucent tape.
Try this with a low-power scope (up to 3X). With your rifle at your waist, pick a clearly defined target. Focus on it. Now shoulder the rifle quickly without shifting or relaxing your focus. The crosswire should appear on target. Your subordinate eye will see past the scope as the reticle jumps into the target plane.
High-power scopes give shooters more trouble because the target image in front of one eye is so much larger than that in front of the other eye. Also, magnified wobble, a restricted field and shorter eye relief delay target acquisition.
The result is similar to what you get shifting focus from a speeding teal to your shotgun barrel. That is, you'll work harder to hit less often. While the rifleman often has more time than a shotgunner, quick accurate shooting is almost always better than slow accurate shooting.
You can learn to use even high-power target scopes with both eyes open. The 20X Redfield on my smallbore prone rifle presents my dominant eye with an image starkly different than what my left eye perceives. But practice has enabled me to shoot with only a slight narrowing of my left eye--and that mainly to reduce sunny-day glare spared my dominant eye by the scope's eyepiece.
Open sights can be more difficult, as you're lining up three images in three focal planes. A double image of the front sight can delay a shot or cause a miss. An aperture rear sight absolves you of rear-sight focus, so shooting with both eyes is easier. Concentrate on your point of aim and center it with the bead. An aperture rear sight lets you aim a rifle much like you'd point a shotgun.
Optically a scope has an edge over iron sights because the reticle is in the target plane and will be seen in sharp focus. Your eye cannot bring a front sight and a distant target simultaneously into sharp focus.
Binocular vision is a marvelous gift. No matter how far you must look or how precisely you must aim, two eyes are truly better than one.
Some of these same principles apply to glassing for game. When I hunted Coues deer with David Miller a decade ago, he carried 30x80 binoculars, along with a husky tripod. He hauled both into the desert hills without complaint because after he sat down to glass, he'd peer through those lenses for hours on end.
Dave has since replaced his 30x80 with two spotting scopes mounted side by side on a special platform. Another option is the 120mm spotting scope now available from Rigby. It features a single barrel and objective unit but with an ocular assembly comprising two eyepieces that can be focused individually. A separate focus dial on the main barrel changes focus for both at once. While big and heavy, is easier to transport and use than two spotting scopes.
For backcountry hunters who must stay mobile, spotting scope magnification is available in some binoculars. Zeiss and Nikon have built 20X and 16X glasses with stabilizers to reduce the image shake that afflicts powerful binoculars when supported only by hand. Leica's Duovid 10+15x50 binocular delivers a choice of 10X and 15X magnification. Doublers, like the detachable eyepieces by Brunton and Swarovski boost power without adding bulk or weight until you use them.