Are the contents of discussion between a University and a student private or public? I have the opportunity to reject an offer from a University in US. As a follow up email I was asked which university did I choose and what was the reason behind it. Can I simply ignore the follow up mail, or can I divulge the information as a good  gesture? 

Personally, I do not like to reveal stuff, but is the information where I am going to study or whether or not I received funding from a particular university public? If I reveal it to staff of another university  will it not be a problem?
You're not obligated to tell them. Whether you tell them or not would be unlikely to have an impact on your future either. 

They're probably not going to track you down and find you, and would mostly use the information to improve recruitment in the future. 

Imagine if you told them you chose University X instead because you had family there and it was easier to secure housing. That's a sound reason, and unlikely to cause any sort of mark in the off chance they were tracking you.

I would try to avoid saying anything bad about the school, though.