If you run an executable stored on a mapped network drive, it runs on the CPU of the computer that mapped the network drive (which presumably you're sitting at), not on the computer hosting the files. In this case, the executable file itself it transferred over the network, and any data files it needs to access.
If you use something like remote desktop, it runs on the computer hosting the the desktop (and the files), not on the computer you're sitting at. The executable and associated data files don't need to be transferred, but all graphics and commands need to be sent over the network.
Auto-save will likely save on the computer running the EXE. Mind you, the default location might then potentially be on a mapped network drive, thus sending it back over the network. This would be program specific though. If it's storing temp files alongside your user settings, then things will end up on the computer whose CPU is in use.
Other strange setups may have custom behavior.