How can you remove another car's paint from the bumper of your car?

Contact you local automotive paint store. The should have the correct products. The major companines are 3M and Mequiers. "Goo Gone" works great! Or, Mr. Clean's Magic Eraser used VERY lightly.

IF its just the paint , try 1500 grit wet sanding , then get a buffing/polish compout for the cars original paint for after you sanded off the foreign paint

So yes - if it's just the other car's paint that is on your car, you can get it buffed out. I'm speaking from prior experience. I've also hit a yellow pole, where the pole's paint got onto my bumper and it was easily buffed (polished) out.

If the scratch appears thin and white, it probably hasn't penetrated through the clear coat.If it is body-colored or shows metal, you've got a deeper problem. You should go to an Overspray shop, surely they know what will they do to your car.