Plenty of hockey players are pretty petty/vindictive - the agent could've leaked it. I'm sure he had plenty of people asking him why he waived his NTC from both teams, all it takes is one player to talk.
Don't need to look much further than MAF's agent posting a photo of getting stabbed in the back to tell you how these people think. McDonagh is 33 and has 4 years left on his contract, he doesn't have to worry about the next contract so it really doesn't hurt him.
Edit - Also this is no disrespect to Nashville, but why would McDonagh waive his NTC? He basically had everything in Tampa - playing on one of the best teams in the league, won B2B cups, low taxes, great weather, etc. From all accounts he doesn't have any ties or connection to Nashville, so why would he waive? Because he's a nice guy?