Maybe I missed this somewhere, but why don't they ask people who spread these rumors, or reveal that they know about these rumors, where they got their info from? I can't imagine none of the people who spread these rumors would be unwilling to help them once they know the information they're spreading is false. Backtracking to the source shouldn't be too difficult.
Also, how does the source keep their reputation intact? How come people believe their gossip when it's been denied by 'involved' parties multiple times?