After a little further exploration, I discovered that the requirements for the “On the Map” app are indeed public (but not publicized). You can read them here. Be certain to follow the link to the rubric if you plan on pursuing the Nanodegree. Those documents do not include the API for Udacity user authentication, but a little poking around in GitHub will probably reward you with the information you seek by way of the repositories of prior students’ projects.
Apparently, the “On the Map” app is a Nanodegree-only exercise. That is very disappointing, as I would like to accomplish as much as possible of the Nanodegree prior to signing up for the Nanodegree program (because you are charged monthly for the Nanodegree program until the work is completed). This is a blatant money-grab by Udacity. There is no reason the “On the Map” resources and requirements could not be made available publicly.
Perhaps the Udacity API is not public, but then why expose it to the Nanodegree participants?