The hosts computer controls the AI and transmits the info to the clients.
I don't know if this holds true for LAN, but on ICO the Dev journals make it sound that every computer is simulating the AI, but since each computer gets the exact some commands and generates the exact same random numbers, the AI acts in the same way on each computer. You get desyncs when for some reason this doesn't happen.
Either way, the AI is certainly the biggest thing that desyncs can screw up. Games with only human players can still be effected, but as you noticed its your commands are still read correctly in a desync so its not as obvious.