When you make a purchase on XBLA, it becomes tied to the account the purchased it, and the FIRST Xbox that it was downloaded to. Anyone on the first Xbox can play it as long as they are signed in to Xbox Live. If you install the game on any other Xboxes after the first, only the account that originally bought the game can play it. If you are not signed into Xbox Live, or have installed it to a second Xbox and are not signed into the account that originally purchased the game, it will default to the trial version.
You can find more information on this at Microsoft's website.