I'm 99% sure this is more "off topic" than "news".
You won't hit where you are aiming. You will hit high and to the left.
For long guns (rifle, shotgun) most instructors will have the shooter use the dominant eye and shoulder the weapon on that side (firing with the non-dominant hand).
For hand guns there are workarounds and techniques if you don't want to use the non-dominant hand. But my experience is that instructors teach shooters to use the dominant eye and non-dominant hand.
is this as hard as it sounds?
If you are right hand dominant, but left eye dominant how does that impact your shooting?
If this is for personal/civilian world, you might also consider retro/holographic type sights so you can go with two eyes open.
If it's for the military.... practice, practice, practice (good advice in any case).