I guess @ShadeOW only wanted to understand the dark magic behind this P2P payment feature of GNU Taler. Am I right?
Indeed, Taler is a P2P service despite the existence of the Exchange. The exchange is only there to convert money to tokens and vice versa. Once your wallet contains tokens withdrawn from the bank, you can pay to a merchant. The payment process is extensively described in the Payment processing section of the Merchant Backend API:
To process Taler payments, a merchant must first set up an order with the merchant backend. The order is then claimed by a wallet, and paid by the wallet. The merchant can check the payment status of the order. Once the order is paid, the merchant may (for a limited time) grant refunds on the order.
Thanks for asking anyway, it was a nice opportunity for me to dig in further in the docs and to suggest some improvements to bring to the docs 