Doing this 1 nail at a time, first, pound the nail RIGHT UP INTO THE STUD where it looks like it pretty much 'broke through' the entire sheet of drywall. (Reason being, if the nail is still in the drywall, then the drywall in conjunction with the stud, will 'work' the nail loose again. You see, if the nail is entirely THROUGH the drywall and ONLY 'sitting' in the stud, there isn't any way it can work loose. See what I mean???) Then, after pounding it through with the Nailset, put a screw an inch or 2 above (or below, or both of) the nail. Buy yourself a special tool that you attach to your drill to recess the screw about 1/4". Home Depot has it in the Tool Coral for about $10.
Sure, you COULD pull the nail out if you want but if you have 80-100 like I did, it's not a pleasant option. Besides, you'll make more work of your spackle job. In some cases, where I was looking like I was doing more 'damage' to the drywall than good, I used some needlenose pliers to 'squeeze' the nailhead into a 'V', then I used the nailset to tap it all the way THROUGH the drywall, and into the stud. You can do this too if you're noticing a lot of 'unexpected damage'. Remember, the drywall UNDER the nailhead needs to go SOMEWHERE as you pound it into the stud. So, by crimping it, you dramatically reduce the 'forcing-out' of drywall underneath.