I figured it out on my own, no thanks to all the conflicting information I found elsewhere on the Dish Network web site and hits from search engines.
What I did to get things to work correctly (on the ViP722):
1. Have both remotes with you in front of the receiver.
2. Press the "System Info" button on the receiver's front panel behind the door on the right end of the receiver.
3. Hold down the "SAT" button on the remote until all lights at the top of the remote begin to flash.
4. While those buttons are flashing, select a new address number on the number key (any number between 1 and 15; I chose a single digit number).
5. Then hit the pound key (the "#" button).
6. Then hit the record button. Yes. You MUST press the record button. Contrary to one of the tech notes on the Dish Network web site, the ViP722 receiver will not see the new address without pressing the record button.
7. You'll see the address number change on the Info screen for one remote. That's when you know you really are making progress. But you are NOT done yet. The other remote is still stuck at the old address getting mucked up by the neighbors. Repeat the same sequence for the other remote control.
8. Using the TV2 remote alone for an address change will not apply changes for all remotes. Maybe this works for some receivers. It does NOT work with the ViP722. At least it does not work with mine. Each remote must have its address changed individually.
FWIW, I had some fun doing random channel surfing with my UHF TV2 remote just to see if I could irritate the hell out of what ever person was granting me that favor over the last couple days. Maybe he's on the phone with Dish Network's tech support. I just hope he doesn't select the new remote control address number I selected. He has a 1 in 15 shot.