Otherwise, when playing the same game across different playstation accounts (not psn accounts), you need to level up for split screen separately.
PS3 save games are per-user not per console.
The game is unable to write save data for one account in a manner that another can use it using the save game system, though there are ways to accomplish this if you want to.
The whole point of the levelling system in MW2 is that an individual player earns it, so I'd be pretty sure that levelling anything will be linked to the account not the console.
You might be lucky though :)