No, you do not need to buy two copies. Xbox 360 supports Game Sharing (License Transfer) for digital purchases, allowing you to play your games on a second console in your household.
If you own the disc, simply move the disc between consoles. No license transfer needed. Save files can be moved via USB drive or Cloud Saves (Gold required).
Summary: Buy once digitally → Set Home Console → Play on two consoles (one offline for family, one online for you).