If you have seen the Usage explanation when you download, it says when you download the Xbox Live Arcade game it's availible for every user (gamertag) on the Xbox 360 you downloaded it on, now if you have a memory unit you can transfer your Xbox Live Arcade game to an other Xbox 360, but you need to activate it with
your account to play the full version on the other Xbox 360, beware you can only
move your gamer profile, it's to prevent achievement cheating

Now I kinda got to correct my self, after a day trying on my friends Xbox 360, you are only able to play the full arcade game on his Xbox 360 with you account only.
If you want your friend to play your arcade game, he has to play the game on your hard drive because there it's playable for everyone
Though this thread was transferring Arcade games between Xbox and Xbox 360, but maybe this will help people out on there 360's
"Based on my findings. The content appears not to be signed unique to each console. Moving premium arcade content from one console to another causes no problems at all. Can access the full package of content."
He's wrong, I had the registration key for Bankshot Billiards from my friends Xbox and tried to run it on my Xbox, didn't work because the checksum didn't fit my Xbox, still don't know about the Xbox 360 if that will matter.