When you buy any DLC item on your Xbox, there are two licenses. One is for the console. This means anybody who uses your console has full access and functionality (online play, achievements) to the game be they online or offline. (Obviously they need to be Xbox Live gold to play online). The second license is on your gamertag. So you can use a friend's console and re-download the item. But in this scenario you need to be signed in to Xbox Live for full access.
There is a tool to transfer the console license to another Xbox but this is only allowed once every 12 months. There is no way to transfer the gamertag license.