Except some of the reports come from insiders. Guys like Rapoport and Schefter are always talking to people, i.e., players, front office managers, PR people etc. They're always on the phone talking to someone. And the someone's they talk to want to have that relationship because it makes the someone's feel important (if they're lower level people) and it's a way the higher level people get information out to the public, players and others around the league as a way of testing the waters or stirring up trouble for competitors.
So, let's say, Schefter is talking to a Saints or Cowboys scout, he may ask, "What do you think about Sean Payton leaving? Do you think there's a possibility he may resurface with the Cowboys?"
And the scout, being protected by anonymity and trusting Schefter won't reveal his identity (this is the way Schefter builds trust), starts shooting his mouth off with hypotheticals. Well, if it's just chatter, that becomes a story. You talk to enough people who give the same opinion, and it gets thrown out in cyber world as a speculation piece. The scout gets what he wants by feeling important talking to an NFL insider (yes, a lot of these people are vain). Schefter gets what he wants. And no one loses face because it's all speculative reporting.