Privacy protocols are guidelines intended to allow computation while still protecting the individuals involved.
One issue with this protocol is that the phone number created might not exist.
Another protocol without this issue is to designate an airline, destination and date, and have the first person make a reservation using the name of their secret, then the second person goes and cancels the reservation using the name of their secret.
A simple protocol that does not rely on a human third party involves password changing.
Then the labels will be removed, the cups shuffled, then flipped over to reveal the slips of paper.