Are you required to report employees that are surfing or visiting illegal websites? According to a recent New Jersey case, absolutely! States other than New Jersey will probably follow their lead and rule similarily in other states.
The New Jersey case was filed by a woman who had a 10 year old daughter. The woman had recently married a man who was employed by "Company A." The woman found out that the the man had been taking inappropriate photos of her daughter and posting them on the web while he was at work. She sued the company because she found out that his HR Deparment and IT supervisor were aware of his visits, but chose only to warn him to stop, rather than report him to the proper authorities.
The court ruled that companies that have web monitoring software have a duty to use it.