The receiving end just needs to be able to play video at whatever output settings you use i.e. 1080p @ 60FPS and have suitable video output for the display connected to it.
The latest incarnations of stuff like Steam Remote Play, Moonlight, etc. are good enough to provide an experience equivalent to 60Hz w/ V-Sync gaming - so single player FPS games are doable, maybe not quite ideal but acceptable, but multiplayer twitch FPS they'd leave a lot to be desired.
A LAN that can fully utilise gigabit or 2.5Gbe to be sure would make it work a lot better - it won't need that amount of bandwidth, generally in the range of 10-50Mbit is used, but there are also latency considerations which having a faster network helps to ease especially if there is other traffic on the network.