Closing the non-sighting eye will definitely make it easier for almost anyone to aim more accurately, since there is no need for the brain to compensate for the off-kilter image of the sights coming in from that eye. Yes, most people can learn to shoot quite well with both eyes open with practice.
In defensive shooting, keeping both eyes open while aiming greatly improves situational awareness and allows for depth perception, which are critical factors. Closing one eye is often not smart.
In bullseye shooting, keeping both eyes open prevents eye strain and improves vision. Having one eye closed for an extended period of time creates a "mixed message" in the brain about light conditions -- the closed eye says it's dark and to open the iris of the eyes, the open eye wants the irises closed to match actual light conditions. For this reason, most bullseye shooters have a blinder over their non-sighting eye so that they can keep both eyes open, but only use one eye for sighting. It's like having one eye closed while it's actually open.