They do occur quickly and are often at very short distances, however...
I have debriefed many officers involved in shootings while on duty and seeing the sights to make hits seems to win out.
1. It does depend on training. If you train to use your sights, you'll see them. The example that comes to mind is trading shots in a hostage situation. Bg's shots when high, LEO saw his sights and made hits while moving.
2. There is a lot of missing when you don't see your sights. Besides the famous NYPD sidewalk shooting, we had one locally where the BG and LEO traded a number of shots across a parked car in a stripmall parking lot. The officer went through half her magazine before remembering to look at her sights and making the stopping hit. Even then, she jerked the trigger and her shot went low...into his thigh.
3. Real close works too. I had an officer who remembers his whole slide being silhouetted by the BG's body. He put all three shots...he though he only fired 2...into COM