I see this question a lot and I've tried to provide answers on many occasions.
First of all, if your pistol has adjustable sights, there's a good chance you'll have to adjust the sights to get it to shoot where you want. It might be counterintuitive, but it's been my experience that an adjustable sight pistol is much less likely than a fixed sighted pistol to be sighted in when it comes out of the box.
If your handgun has fixed sights, then where it shoots depends on a few things, but,
in general, it's been my experience that
most fixed sighted pistols have the sights set more or less the same unless somebody screwed up at the factory.
They will usually be sighted to put the bullets directly on top of the front sight at 3 to 5 yards, a couple of inches above the front sight at 15 yards (suitable for a center hit on a 4" bullseye with a 6 O'Clock target hold), and about 4" above the front sight at 25 yards (suitable for a center hit on an 8" bullseye with a 6 O'Clock target hold).
Here's a plot that I made by collecting the data from targets shot at various distances today at the range. I shot the targets using an STI-GP6 and American Eagle 115gr 9mm ammunition.
The point of aim is the point on the target that is immediately above the front sight with the sights aligned properly.
I shot groups at 1, 3, 7, 15 and 25 yards and measured the elevation of each shot with respect to the point of aim. Then I averaged those elevations to come up with a single elevation figure for each range. The elevation figure at 0 yards is the distance from the top of the sights to the center of the bore since that's where the bullet is, with respect to the point of aim, the instant it exits the muzzle.
Note that since the barrel is below the sights, at the muzzle and at very close ranges, the bullet will actually impact BELOW the top of the front sight.
Because people want to be able to hit objects at farther distances, the bore has to be angled up very slightly with respect to the sights. If that weren't true, given the effects of gravity, the bullet would never rise to the line of the sights since it obviously starts out below the sights.
So the answer to the question that titles this thread is:
It depends.
No matter how your pistol is sighted, it will probably shoot to significantly different elevations at different ranges.
However, one thing should be clear. It doesn't make sense for a manufacturer (or gun owner) to sight a pistol so that it hits significantly below point of aim at distances like 7 to 10 yards. It's become common for people to claim that certain guns are sighted this way, however not only has it not been my experience that such a thing is true, it doesn't make sense.
One can look at the plot and see that if the line angles downward, instead of upward, it will keep angling downward, and it will be closest to being on target at the muzzle--dropping farther down on the target as the targets get farther away.
If one were to sight a pistol with an initial sight height of 0.5" so that it hits 2-3 inches low at 7 yards (behind the front sight), we can calculate that it would have to hit the target about 4"-6" low at 15 yards and it would be 6"-9" low at 25 yards.
[Actually it will be lower than that because without the slight upward angle to "fight" gravity, the bullet's initial downward trajectory and gravity will work together to drive the point of impact even lower than these calculations show. The calculations assume a more or less linear trajectory which works well for a close range (inside 25 yards) approximation of a bullet trajectory with a typical slight upward angle at the bore. They will underestimate the effects of gravity if the bore is angled downward as it would have to be to achieve a point of impact at 7 yards that is significantly below the point of aim.]
When sighted like my GP6 (and most of my other fixed sight pistols) came from the factory, the bullet will only hit lower than the point of aim at very close distances. It must because the bore is below the sights, but, even then it won't hit very low as the shooting results in the plot show. The distance between the top of the sight and the bore centerline is very small--less than an inch, so it won't ever be lower than that until you get way out past typical pistol distances and the trajectory bends back downward due to the effects of gravity.