When you book flights through Duffel, it's your responsibility to send a confirmation email to the customer with the itinerary, payment details and booking reference (which the passenger will use to check in). The API should include all of the details you need to send that email. Duffel will never contact your customer directly.
At the moment, we don't offer an option where Duffel sends the email instead. If this is something you'd be interested in, please do get in touch with us.
A small number of airlines - mostly "low cost carriers" - will also send the passenger their own booking confirmation with their itinerary, and sometimes the price paid for their booking. You can read more about this here.