It is common practice to polish the chambers of double guns used in Cowboy Action Shooting. This is done so the empties will fall out easily (ejectors are not allowed in CAS). I've used very fine (600 grit) emery cloth. Slotted the end of a 1/2 inch dowel and put strips of emery cloth in so that the grit side would be against the chamber when the dowel was chucked into a 1/2 drill. Finished with polishing compound on a rag, also using the dowel and drill. Don't how you would do that on a pump gun without taking the barrel off. I've done this to three shotguns. It takes a fair amount of time even with the assist from a power drill to make any real difference in the smoothness of the chambers. I wonder if the condition the chamber is really the problem? If so, it seems to me you have a warranty situation with Remington. A new 870 should shuck those empties every time.