Trying to drill out the screws is not a workable solution for the type of electrical box shown in the picture. The plastic or composite material that the box is made of is much softer than the steel screws. Attempting to drill the screw, especially with a hand drill, will just lead to the drill bit skidding off the top of the screw and digging into the adjacent plastic.
The screw that still has some of its threads sticking out may be able to be removed by clamping onto the stub with a small pair of "Vise Grip" type pliers and turning it with the pliers once gripped on.
The problem still exists however with the other screw and even if you are able to remove the one with the stub you are still faced with the need that this electrical box will have to be replaced with a new one.
There are a variety of electrical box types called old-work boxes that clamp to the wall board and can be used as a replacement as long as you can get the old box out of the way. That old box is likely nailed to a stud on one side or the other. Removal usually involves removing some wall board and then patching up once the new box is in place. However it is sometimes possible to use one of those hack saws where the blade sticks out the end. You can use it to saw between the side of the stud and the old box to cut off the nails that mount it in place. If done carefully you may just be lucky enough to be able to remount an old-work box in the same hole.
Another replacement alternative is to mount a new double wide box and then provide a pair of the dual outlets. With careful work you can enlarge the opening just right for the double wide box to fit. This can give you room to work on the removal of the old box even if that means breaking it apart into pieces. Note that there are some clever modern electrical box designs that can mount to the side of the stud using screws that are accessed at an angle from the inside area of the box. Some of these boxes are even adjustable to get just the right wall surface match with the face of the electrical box.
Here is an example of one of the screw mount dual width boxes: