Personally, I like to deliver the dog and help make sure everyone settles in together, and do the adoption paperwork then. It's the last stage of the process, a final check to make sure everyone's happy -- after meet-and-greets, vet and personal reference checks, and all the other screening is done.
It allows me to make sure if there's an existing dog, we do a good re-intro instead of the adopter just marching in the house with the new dog. Even though the dogs have met at least once before, in order to approve the adoption, we still want it to go smoothly when the new dog arrives, so we like to help with it.
While there, we go over all the things to expect the first couple of nights, crate placement, collars, commands, potty schedule, etc. It's often a long visit, but while we talk, the dog is relaxing and getting acclimated to the smells of the new place, the voices, etc. We also have lots of time to verify that things really are as they seem to be in the application and feel very comfortable about the placement, before we leave the dog there.
As long as the home check is done at some point, delivery of the dog is not a requirement, but adopters seem to like it (as do I).