No. Xbox Live's architecture requires each player in a public online match to have a unique Xbox Live account on a separate console. Split-screen functionality on Xbox 360 is restricted to local offline modes only—campaign co-op, private matches, or system link.
To play Call of Duty: Modern Warfare (2007) or Modern Warfare 2 (2009) online with a friend on the same couch, you need:
Split-screen is available for offline multiplayer (private matches against bots or friends on the same console) and campaign co-op in some titles. Check the game's back-of-box features for "2 Players Local" or "4 Players System Link."