When Alice wants to send Bob some coins all she has to do is sign a transfer of ownership with her private keys and publicize that announcement until Bob is happy with the number of witnesses. For small transactions, Bob might be happy with a couple dozen witnesses but with large value transactions he'd want thousands of heavy peers to witnesses the transaction. Why?
Bob's coins only belong to him if someone on the network can provide proof to everyone else, on demand, that those coins belong to Bob. If all of Bob's witnesses go offline then there's no one to vouch for him! In such a scenario, Alice would be able to transfer the ownership of those coins again since no one would be there to stop her.
This is basically turning the current popular algorithms upside down since you're not flooding the network with transaction data, you're listening to the network to make sure no one is trying to take your coins. As long as you yourself are online ready to provide proof, the transfer of ownership signed by Alice, then no one can take your coins away from you. Ideally, you'd have many peers ready to provide that proof, not just you in case you go offline and in exchange you either pay them a small fee or you also host others' proofs ready to serve them up to whomever asks for it.
The only attack I can think of is Alice announces the transfer of ownership of coins to Bob and announces another transfer of ownership of the same coins to Mallory at the same time and hopes that Bob and Mallory don't hear each other. But if Bob and Mallory are smart they'll query enough peers such that their queries overlap and collide in at least one peer on the network.
So to wrap up: using witnesses to 'hear' the transfer of ownership and at least one of those witnesses ready to defend the transfer of ownership is enough to prevent double spends. Caveats are that you want enough witnesses to hear the transfer and you want at least one witness always online. In bitcoin, your funds are safe even if go offline but here your funds are only safe if you're online 24/7. Otherwise you need to trust 3rd parties to defend your coins for you. Are there any attack vectors with this? It sounds pretty simple, too good to be true? Consensus is too slow and we don't need consensus here. And you don't need synchronization since everything would be dynamic.
Can this Proof of Witness be attacked in a way I'm not seeing?