If I received a Christmas present from someone who bought it at Costco but they don't have any gift receipt or receipt how would we go about returning the item?

Simply have the "Return" desk to refund you, either cash or credit back to the account; Costco always has outstanding policy to honor customers' Returns Items. No problem at all. Costco is great!

Take the item to the returns desk. They will have to look up the purchase history of the person who bought it. It may take a while. You can help by having the name, address, phone number, costco member number, etc of the person who bought the item.