Some people get the offer in their acceptance packet, while others get the offer on a later date, separate from the acceptance letter (e.g. get acceptance in March and get the scholarship offer in May). At least that's how it worked a few years ago.
From what I know about those who are on the full-ride scholarship, you have to be someone who has a potential to make significant contributions to the school as a student and as an alumni. It's not something that is guaranteed for someone with a 4.0 undergrad and graduate GPA or 27 on DAT. My impression is that the selection is based upon the entire application package, and candidates are evaluated holistically.
A friend of mine got a half-tuition scholarship after D1 (i.e. he had to pay half the tuition for D2-D4 years). I believe he was the only one in my class who was offered this particular scholarship. He did some significant leadership activities.