which bumper? how deep is the scrape? is there body damage? do you have contact info, plate#, or insurance info for offending vehicle? was it a hit and run?
if its just a scuff in the paint, i would recoment rubbing compound (fine grit) and a good polish/wax.
if it is a deep scrape (to the plastic/metal) you may have to have the area resprayed. it will be easy to blend the new and old paint if your car is a non metallic color (red or white non pearlesent), but it is still possible to get it blended (mine turned out really nice and i have black mica).
if there is body damage you may need a new bumper plus paint. i would go through insurance for this (mine was ~$2000.00 for a new fender and paint from a really nice shop in town. i also got a free wash and wax every 2 months for life

went through her insurance too b/c it was a hit and run).
i would get an estimate to get it fixed in a manner that you will be happy with from a good body shop, and do it before the offender reports it preferably.
b/c it seems to be their fault, their insurance company should deal with it. if you dont have any info from them you will have to go through yours. if it was a hit and run make a police report and report the plate # to your insurance co, they will deal with getting reimberssed.
the other option is just to live with it if its not that bad.
i just went through a smashed fender on a hit and run in a parking lot a few months ago, so i know how you feel
-Nick