So there's not really a secret element to this, if you e-mail the person you get and say "hey, I have you... what's your address." lol
Doesn't the person in charge usually collect everyone's names and addresses, and then privately send a name and address to each person who signs up? That way the person sending you something is kept a secret.
Also, typically the person you receive something from is different than the person you send something to. Basically, I think you create a list of everyone who wants to participate... and then you put all of the names in a bowl and go down the list drawing a name for each person. That way you don't end up with people waiting on their "partner" to send their gift first... or people upset because the person they sent something to didn't send something (in their opinion) as good or as expensive.
I know that all sounds petty... but I've taught long enough, and done enough gift exchanges at Christmas time to know how people can be with this kind of thing.
Jim