If you buy something off the marketplace there are two licenses, one tied to the console and the other to your gamertag. So if you were to copy an arcade title and your gamertag to a memory card and take it over to a friend's place, you could play the full version of the game (providing you could log into Live and that it was the gamertag that purchased the game) and somebody at home could play the full version of the game (as it was also licensed to the console).
Things went crappy if you had a console replaced, as you would lose the console license and only have the gamertag one, so you always had to log into Live to play the full version and other family members could not play the full version, unless the gamertag that purchased the game was logged in.
They fixed that a year ago with the license transfer feature that allows you to transfer all console licenses or combine console licenses (if you own more than one) to another console. The limitation is that you're only allowed to do it once a year, I guess to prevent someone transferring their licenses to a friend's console for a weekend.
I think Blue's comment is regarding the "Install to Hard Drive" feature that the NXE brought in. I imagine the install is tied to the console ID, hence the need to reinstall the game disc. Also, I believe he may need to redownload any Arcade titles he purchased if other gamertags want to play the full version when the gamertag that purchased the game is offline. A quick check would be to login to Live using a different gamertag other than the one that purchased the content and see if you have the full or trial versions of the games. If it's trial, you'll need to redownload the games.