You will receive your letters and mail at the address you provide, sent out on a schedule provided by the selected service or experience.
Due to the different speeds at which the mail service works, depending on your location and distance, it may take as little as a few days or up to a week from the date of sending until you receive your letters.
If you choose not to respond to your mystery partner or even simply forget, they will proceed with their investigation as events unfold and update you with their next scheduled letter to let you know how things are going.