Dealing with Latency in Networked Games
I find this network physics blog post by Glenn Fiedler, and even more so the response/discussion below it, awesome. It is quite lengthy, but worth-while.
In summary
Server cannot keep up with reiterating simulation whenever client input is received in a modern game physics simulation (i.e. vehicles or rigid body dynamics). Therefore the server orders all clients latency+jitter (time) ahead of server so that all incomming packets come in JIT before the server needs 'em.
He also gives an outline of how to handle the type of ownership you are asking for. The slides he showed on GDC are awesome!
On cheating
Mr Fiedler himself (and others) state that this algorithm suffers from not being very cheat-proof. This is not true. This algorithm is no less easy or hard to exploit than traditional client/server prediction (see article regarding traditional client/server prediction in @CD Sanchez' answer).
To be absolutely clear: the server is not easier to cheat simply because it receives network physical positioning just in time (rather than x milliseconds late as in traditional prediction). The clients are not affected at all, since they all receive the positional information of their opponents with the exact same latency as in traditional prediction.
No matter which algorithm you pick, you may want to add cheat-protection if you're releasing a major title. If you are, I suggest adding encryption against stooge bots (for instance an XOR stream cipher where the "keystream is generated by a pseudo-random number generator") and simple sanity checks against cracks. Some developers also implement algorithms to check that the binaries are intact (to reduce risk of cracking) or to ensure that the user isn't running a debugger (to reduce risk of a crack being developed), but those are more debatable.
If you're just making a smaller indie game, that may only be played by some few thousand players, don't bother implementing any anti-cheat algorithms until 1) you need them; or 2) the user base grows.
we have implemented a multiplayer snake game based on a mandatory server and remote players that make predictions. Every 150ms (in most cases) the server sends back a message containing all the consolidated movements sent by each remote player. If remote client movements arrive late to the server, he discards them. The client the will replay last movement.
Check out how Valve does it in the Source Engine: http://developer.valvesoftware.com/wiki/Source_Multiplayer_Networking
If it's for a first person shooter you'll probably have to delve into some of the topics they mention such as: prediction, compensation, and interpolation.