I know that when you cast someone else's spell, all of the "you" and "your" affects apply to the person whose turn it is. I assume the spell gets discarded as it normally would (unless it says otherwise), but the instructions say to put it in "your" discard pile. I'm guessing it still goes into the card owner's discard pile, but I wanted to check.
Indeed, it goes to the discard pile for whoever's breach it was prepped in. The 'you' is only the 'you' on the effects of the card.