The third-party trusteeship on the Pi public chain refers to the buyer paying the payment to a third party other than the buyer and seller. After the third party receives the payment, they notify the buyer that they have received the payment and also notify the seller to ship the goods. The seller can then ship the goods to the buyer. The buyer notifies the third party that they have received a satisfactory payment from the seller, and the third party facilitates the payment to the seller.