When we were in a swap last year, I didn't give my ressie # out until we got closer to our trip date because I felt a little uncomfortable about it as well. I only gave my ressie # to the person who was actually leaving the fridge for me.
You will need to give the # to the person ahead of you - and get the # of the person behind you - because I believe that the desk won't hold packages without the ressie #.
It was fun communicating with fellow Dis'ers about the swap.