Changing the wake on LAN setting didn't help.
I don't think this can be a surprise, as wake-on-LAN is intended to power on the machine, not kill the screen saver or take the monitor out of power-saving mode. The other machine works fine without wake-on-LAN set.
Maybe my first post wasn't clear. The machine in question is powered-on. It's screen is in power-save mode, since it's not being accessed. I expect it to work like the first machine: the monitor enters power save after a time, but if I use Multiplicty to control the machine, it should wake up again.
Other mouse-sharing programs don't have this problem on this machine.