Both players reveal their hands and add 1 card from each other's hand to their hand.
Rulings
Individual Card FAQs
You cannot activate "Exchange" if you have no other card in your hand besides "Exchange".
The opponent is still considered the owner of the card you take by the effect of "Exchange" for all purposes.
If you take your opponent's "Kuriboh" through the effect of "Exchange" you may discard it to activate its effect (but it is discarded to your opponent's Graveyard).