The conversation is NOT private when it happens during a school event. 
Osborn was at a Creswell vs. P Hill sporting event in the role of athletic director when he decided to make disparaging remarks about the two young ladies. 
Bracco was sitting in his office at CHS and knew he was being filmed when he decided to display his middle fingers for the camera. 
IF this had happened off campus, in their homes, at a park, anywhere that wasn't connected to CHS, then I would agree it was a "private" conversation. Not a nice conversation but a private one. 
When you are representing a school your behavior has to be beyond reproach. 
I am very disappointed it took Hamilton so long to take action. I have a hard time with his continuing to pay them pending the outcome of the investigation. Unfortunately, he doesn't know how I feel as he hasn't returned any of my phone calls!!