After stirring paint using my drill and thus making a big mess outside on both the sidewalk and grass, I figured out how to stir paint without making a huge mess.
Step 1. Find a paint can lid.
Step 2. Use a 5/16" metal drill bit to drill a hole in the center of the paint can lid. I used a drill press as it gave me the leverage I needed.
Step 3. After drilling the hole, you're left with a metal burr. I used a sharpening block to sand the burr down but sand paper will work as well. Be careful as the burr is sharp and in fact, may cut through the sandpaper if applied by hand so you might want to use a sanding block instead.
Step 4. Place stirring stick through the hole to make sure it fits snugly. You don't want it to be too loose as paint will come through the hole and make a mess, which is what we are trying to avoid.
Now you should be able to place the stirring stick through the hole and place inside the paint can. Slide the lid down the stirrer and hammer onto the paint can to be stirred. I tested this out by placing my drill to high speed setting and although a little paint did come through the space between the stir stick and the hole, it didn't splash around and was easily wiped off. Of course, you don't need to use high speed to stir paint.