Discarding your hand is not a cost for either player.
Netrep Rulings:
If there are any cards that are not in the owner’s hands, through the effect of “Exchange” or similar, when “Card Destruction” is played, the owners of the respective cards may decide the order that the cards go into the Graveyard.
Cards discarded for the purpose of “Card Destruction” are not a cost of activating “Card Destruction.”
If “Card Destruction” resolves while an active “Banisher of the Light” is on the field, each player still draws the same number of cards that were discarded.
“Card Destruction” can be activated if either player has at least 1 card in his/her hand (not including “Card Destruction”). If your opponent has 1 or more cards in his/her hand and you only have “Card Destruction” in your hand, you can activate “Card Destruction”. If your opponent has no cards in his/her hand and you have “Card Destruction” and 1 other card in your hand, you can activate “Card Destruction”. If your opponent has no cards in his/her hand and you only have “Card Destruction” in your hand, you cannot activate “Card Destruction”.