There may be many reasons you want your scholarship/discount information to remain private. Perhaps you got a great deal, and you don't want your roommate to feel bad if s/he didn't? Perhaps you don't want people to know you paid full price or paid zero, thereby exposing that you are at one end of the wealth scale?
Releasing only anonymised data may help with concerns for individual privacy. Degree-granting institutions are licensed by the government, so perhaps you can argue that releasing anonymised data on this sort is a fair trade for being granted that privilege.
That said, we don't need super personal data. Summary statistics would be fine.
On that note, the NPR article cites this survey[1], wherein participating private colleges contributed data that was then used to calculate aggregate statistics about students in the US in general.
What we'd want is probably more specific than that. We need per-school statistics, so that trends in how colleges charge and discount may be identified (what if colleges on the East coast are more likely to mark down tuition? What if advertised tuitions are relatively constant, but actual tuition paid varies with cost of living? these are interesting threads to explore). The difference between the average student's tuition after grants and the sticker price, along with the standard deviation of that distribution, would be a good place to start. Quantiles would be even better.
[1] http://www.nacubo.org/Documents/about/pressreleases/2013TDSP...