It looks like xsplit has a local streaming feature, so if you're willing to pay this might work for you. Try before you buy, I have no experience, just found it while researching RTMP servers.
If you want to do it for free, you need three things - a broadcaster, a media server, and a viewer.
So, for instance, you might run
OBS on your computer (the game playing PC), then an RTMP server on another computer, then a media player on the viewing computer(s). The RTMP server may end up on the viewer, so you don't necessarily need three computers.
RTMP servers are plentiful on linux, so if you're comfortable with that then you may be in luck. If not, there are RTMP servers available for windows, but the little looking I've done suggests they can be hard to work with, and don't play nice with every broadcaster and every client. Could be very fiddly.
Clients abound - most media players will play an RTMP stream. Once your RTMP server is set up it should provide a URL to go to, and when entered in your browser it should pull up your default streaming app, if not simply play it in the browser.
Open Broadcaster Software (OBS) is free, and designed to stream to a variety of streaming services, but you should be able to get it set up to stream to a server you choose on the local network.
Here's a bunch of links that each have some interesting information, but none of them provide a total solution. Perhaps they'll give you a few paths to try out, though:
http://stackoverflow.com/questions/...al-server-which-can-convert-the-stream-into-a
https://obsproject.com/forum/threads/send-obs-stream-to-another-obs-client-through-internet.33283/
https://obsproject.com/forum/threads/locally-streaming-is-this-possible.1570/
https://www.reddit.com/r/leagueofle...ltimate_beginners_guide_to_streaming_with_obs
Google search for "windows rtmp server"
Looks like NGINX is the preferred linxu RTMP server.
Let us know what you learn and find as you go along, this is an interesting topic.