There are two elements of what you've written that stand out to me:
Element One: You've had several colleagues inside and outside of your workplace telling you about a gossiping co-worker, and the negative effects that's having with your subordinates.
Element Two: You've also noticed that your subordinates acting more disrespectful of your authority as time goes by.
Re: Element Two – This would be the more substantive issue of the two, so hit this one head on by sitting down with each disrespectful subordinate and simply stating that showing you, their supervisor, proper (more) respect is a "formal requirement of their job." As such, future infractions will be dealt with in accordance with company performance management practices, which can lead up to, and potentially even include, termination. Let them know, as well, that whether they choose to show you more respect, or not, is entirely their choice -- you just wanted to "formally inform" them about the potential consequences of their actions. What's absolutely lovely about this approach is that you don't need to get wound up about it; it's a matter-of-fact piece of information that you wanted to be sure to share with them so that they understood. After all, that IS the respectful thing to do. Tell them, as well, that you will be preparing a memo that "formally documents" your discussion (along with any response and/or commitment they'd like you to include for them) for their permanent employee files and upper management's information. Then thank them for their time and pat yourself on the back for a job well done!
Re: Element One – So lemme get this straight...you learned, from gossiping with others, that the person you were gossiping about has been gossiping about you?! [Pause for effect.] Let it go, my friend. If you handle Element Two the way I suggested, that gossiper will soon have no one to gossip to.
