I must respectfully disagree. I shoot now with my left eye closed, RH due to being left eye dominate. I do see the arrow in my peripheral vision, and I use it to line up the shot. The distance "judging" is accomplished with both eyes open, I draw, close my left eye, line the arrow up (split vision is probably the correct term here) and release. Granted you do have to practice this quite alot to get used to different distances and angles. I've tried to shoot instictive this way also, I can, but I tend to hurry my shots, most likely due to telling myself not to look for the arrow. Nothing against insticntive shooting, started that way myself, but started fooling around with the split-vision method, found out about my left eye being dominate and started to shoot with the eye closed.
As the saying goes, different strokes for different folks. In case I get flanked for shooting that way, I've always shot rifles and pistols with one eye closed. Did well enough in the military to qualify as expert with both, even shooting the pistol left handed.