I still shoot with one eye closed. Both eyes open slows me down, with the same "two sets of sights" issue.
You can do a search here on eye dominance (or eye dominant, or cross dominant) and see what pops up.
Eye dominance: some folks have a strongly dominant eye, and that eye "takes over" with both eyes open so they have no problem shooting that way. Others have a weakly dominant eye, or even co-dominance. For those people, there's often no dependable way around the "two rear sights and two targets" problem when focused on the front sight, or the "two front sights" problem if focused at the target, other than to close one eye.
Tactically, open your eye immediately after the shot, as you scan and search for other threats. For those who can and do keep both eyes open, great.
Some people are successful in training themselves to shoot well with both eyes open even if it started out difficult. You can try that. But it didn't work for me, and it might not work for you. Everyone is different, and training will not work for everyone.