Most people shoot at "center mass" somewhere in the center of the upper chest. "Between the nipples" is a classic area to try to shoot in silhouetes and in reality, should be a heart shot. Chances are, as noted, you will be lucky to actually hit where you want to be aiming. A little left or right still produces a solid hit, quite likely a major blood vessel or lung shot. A little high or low will still be hurtful in the throat or in the liver area.
Center mass is actually a bit lower than between the nipples, more along the lines of being below the sternum (around the area of the liver). At that central point, you have the greatest range for error while still producing a hit on the target. If you go low, you end up with a gut shot. High results in a good upper chest shot. Left and right have the possibility of hitting a variety of organs.
So a "center mass" aimed shot really has the best chance of doing significant damage although it may not hit where you want it to hit. This will be due to the fact that the target is moving, turning, and possibly altering elevation (i.e. ducking), and as mentioned, your fine motor skills probably won't kick in as much as hoped for.
Just as a note, when doing silhouetes, I tend to shot high chest (between nipples) area as well, although this may not be the most situation strategic area for a real fight.