thing is , i don't know how it is in the US military / swat / special forces but we in europe ALL learn to shoot right handed..even lefty's will have to learn to shoot right handed.
there's 1 major thing why..and that's the gasmask we use overseas has a cartridge on the left side... so shooting becomes nearly impossible...well it's possible , but you ain't gonna hit anything cause you can't see the sights

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i used to have my marksman/sniper certificate when i was in the military , and although my left eye is dominant , practice like many people say will greatly improve your less dominant eye. lots of people have a twitch before they squeeze they trigger, to counter act that they aim different.
I see it all the time at the range , a little flinch at 150 yards can make all the difference in a miss or hit.
when i squeeze the trigger i NEVER know the exact split second the gun is going off..i know when i start pulling it's going to go off, but i can never pin point it, as i don't just pull the trigger i sqeeze it.(makes sense?) this will greatly reduce twiches you may or may not have... try it out on the range next time
One test that has been purported to show "eye dominance" is as follows:
1) With both eyes open, look at an object in the distance.
2) Raise a finger and line it up so that your finger is overlapping the
object.
3) Now close one eye. If it looks like your finger "jumped" (it is no
longer lined up with the object), then the eye you closed is "dominant."
This test works because your two eyes have a slightly different
perspective, and the "dominant" eye was used to line up your finger.
True , however lots and lots of practice will improve your non dominant eye , and when you no longer have a substantial difference in your dominant eye and your non dominant eye , you will see that is becomes very difficult to line up one finger to an object in the distance, atleast that's the way it is for me..