no, it aint a separate troop model, it's one of your list's troop models
"Each player will give his opponent 1 troop model form your army list to represent the traitor"
since it is not defined as to *when* this happens, the question of whether or not a 10-man SM squad still remains 10 strong at the start of the game is very valid. Basically, the question is: does the exchange of troop models happen at the start of the game or before the start of the game? If it was before, then the answer is no, but if it is at the start of the game, then I believe you'd be able to choose one or the other.
*edit* and yes, with that hideous wording, two opponents can control the same piece of terrain..... good god GW, try and have some self respect