Even in an online FPS all your inputs need to be sent to a server running the game logic so what you're seeing is not an actual state of the game but rather the predicted one. I think it would make sense to have a competetive game run all it's logic and rendering on a single machine for all the players in a given match so that it works a bit like a split-screen - the only difference being that instead of showing it on a single combined screen it would send those screens to remote players. This way a lot of problems with latency and movement prediction would be gone. Also cheating would be impossible which is a good enough reason to do it on it's own.