As a cross eye/hand dominant shooter, but not a 1000 yd shooter, I'll add my 1 1/2 cents. I am naturally right handed however when trying to teach me to shoot at an early age, my father noticed I was trying to use my left eye while shouldering right handed, so he switched me to left side. (Both he and my older brother are left/left so I'm thinking that had something to do with it also). Everything was good and I shot rifle, shotgun and archery well left handed. We only had long guns growing up and I bought my first handgun at 18. Guess what? Shooting it with my non dominant hand felt awkward so I shoot handguns from my right side, right eyed and I feel I'm a decent shot. Somewhere along the way, after reading many eye dominant issues among shooters, I tested myself and found that I am actually right eye dominant. I tried switching to the right side but it never felt natural so I finally said screw it and go with what feels right. I can shoot from both sides but when hunting or trying to shoot the smallest groups I'll go with what feels natural, lefty.
Again I'm not a 1000 yd shooter, but information I've gleaned from those that are, suggest mechanics and consistency are very important. My opinion is that with high magnification scopes and distant targets the cross eyed/hand issue is mitigated, unlike say, trap or skeet shooting. I'd say go with what gives you the most comfortable and consistent trigger control and body/head position behind the rifle. Build your rifle and try it both ways and go with what works. Since this is deliberate shooting I don't think it really matters which side the bolt's on. From prone or off a bench, I'd rather have the opposite hand action to how I'm shooting. For hunting or where action manipulation from different positions may be necessary, matching action to hand is mo better.
JMO YMMV
