I lucked into learning to shoot with a small weekly local group that included a future many time US National Champion, and another person who was in the top five nationwide.
Competitive pistol shooters actually use several different sight picture styles.
In the speed styles of competitive shooting, the goal is to hit targets as fast as possible, so you want to make each shot in the "worst" way that will give you about a 95% chance of a hit. So for a close, low risk target, a shooter may look only at the target and ignore the sights, for a tiniest fraction more speed.
For most targets, the looking at the front sight is correct. Shooters tend to lock their upper body into one shape, then pivot it from target to target while shooting a string. This locks the rear sight in just the right place behind the front one. When the front sight is put on target, the rear sight is automatically in the right place. It's true that the target does become blurred a little when you do this.
Then for really far targets, you do have to bring your focus back a little farther, and see and care about both sights.
The sight picture is not the only thing that changes from target to target. You usually budget the amount of time spent for each shot.
Surprisingly, many pros know where their round will hit before it reaches the target. The time penalty for missing a shot is so high that it's almost always better to take a second shot in case of a miss. However, it takes a while for a pistol shot to reach the target, and for your eyes to see where it landed (plus you'd have to change your focus to look for it, then back again to your sights). To get around that, with practice, you can know in the moment you pull the trigger where the round went, and follow it up in about a twentieth of a second with another round.
In most competitive pistol matches, the sequence of targets to be shot on a given stage is not rigidly defined. There are often plenty of constraints (this group must be shot before these) or timing related constraints in some sports (shooting this target will cause a pair of targets to pop up in 1.2 seconds). Given this, there's a surprising amount of planning that goes into discovering the optimum run. The details of each shot are then worked out and mentally rehearsed.