The university should take the advice of Richard Spahr, and re-learn the art of peaceful verbal confrontation. First it should be ascertained whether the students made the remarks. If they deny them, the other party should ask if they would be so kind as to repudiate and denounce the insults in question, so as to assuage the doubt and anger which plague the community. Otherwise, they should be challenged to either defend their conduct or apologize fully for it. I seriously think that such a process, if people cared enough to pay attention to it, would to a lot to heal the community of the strains caused by racism.
Healing is usually a painful procedure, of course.
We cannot always get inside other people's heads, but the only way we can know whether we have persuaded them is if they remain free to disagree. We have also seen in the past few days that when such a case is resolved secretly, the public can end up suspecting the worst of everyone involved. If the accused racists are given a forum in which to make a public, practically irrevocable apology, most people's minds will be more settled. If they continue to defend their racism, or refuse to show up for an equal debate, we can conclude that we have done all we can for them and renew our efforts to persuade others. As for their privacy rights, I doubt they could claim privacy for anything shouted in a public place.