Under the general heading of "double-tap" there are a lot of different terms used a bit interchangeably for a couple of different procedures. "Double-Taps," "Hammer Pairs," and probably others.
One procedure is to acquire a sight picture, fire, reacquire the sight picture, and fire again.
The second procedure is to acquire the sight picture and fire twice -- with no more time involved than needed to reset the trigger. If you are practiced at this and have the cadence down that allows you and your particular gun, and the power/recoil of your ammo, to cooperate, your second shot will break as the gun is coming back to the original point of aim and your two shots will be very close to each other on the target.
Some shooters describe following the front sight with their eyes the whole time. I can't do that. I find that I see the front sight arrive at the point of aim, fire and then maintain that stance and focus as I look "through" the gun at the point of aim for subsequent shots. My trigger finger is moving and the gun is moving, but my stance and focus are not. Once that first shot breaks, I see the gun as a dark blur, centered on the target.
If you practice "Bill Drills" (6 shots on target at maximum effective speed), you will see this more fully. I cannot say that no shooters execute "Bill Drills" by obtaining a clear sight picture between every shot. I can say that I certainly don't. I generally produce a sub-5 or 6" group in a "Bill Drill." That's can be all "Down 0" hits, if my point of aim and initial sight picture was good and I can maintain focus. I'm not saying that I shoot perfect scores all the time...or much
.. but dropping a "Down 3" is unusual for me with this technique, and it gets me around 0.20 sec. split times, so I'm happy enough. Again, 10 yds. max.
At 10 yds. and under, this is the type of shooting I do in IDPA competition. At longer ranges, sights for every shot.
I've heard of a practice drill some folks use to train themselves to do this. Take one of those small paper plates and try this technique at 5 yds. When you can keep all your double-taps on the plate at 5, move back to 7. Then 10. If you can extend your range beyond that, great, but it's going to take a lot of time.
Good luck! No technique is compatible with every shooter, and vice-versa.
-Sam