I don't know about that - I'm not sure if I ever passed it along to anyone or not after I got it myself (which happened after 3 days of trying and taking my DS everywhere). I just know it is possible to street pass someone without the game on or in your system, but the Eon Ticket may not work that way.
So I guess to be a sure deal, it might be best to do as you suggest and have the game in and on
So I guess to be a sure deal, it might be best to do as you suggest and have the game in and on