Do Both Players Need a Game Copy for Xbox 360 Online Co-op?

Yes, in almost all cases, each player needs their own copy of the game to play together online on Xbox 360.

Exceptions

• Free-to-Play Titles: Games like Happy Wars or World of Tanks allow free downloads, though some content may be locked.
• Xbox Game Pass: If the game is in the Game Pass catalog, subscribers can play without owning a separate copy (requires active subscription).
• "Play Anywhere" / Cross-buy: Rare on 360; mostly an Xbox One/PC feature.

General Rule

For standard retail games (Halo, Call of Duty, Gears of War, etc.), both players must own the game — either physically or digitally — to join each other's sessions via Xbox Live Gold.