Twaryna’s computer will act as the server, and the stream will be on his channel. The plan is to capture the image from computers on the local network, and Twaryna will process it and send it to the internet. I’m not very good at this, so I’m asking for help from experienced streamers on how to organize the image transmission over the local network using your favorite programs.

I’ll try to outline here everything I’ve found; there are several options.
If I understand correctly, they all use Real Time Messaging Protocol.

1. In the paid version of XSplit (like Twaryna has), there is a function for local streaming (if it’s not there, it’s not essential for Twaryna).
Here, in the premium version.
You can install a cracked license (install and move 2 other files to the root directory; if there are glitches, update and move these 2 files again). Download.
The method seems simple, but the forum with a detailed guide is currently unavailable. http://www.xsplit.com/support
(Since Twaryna has a paid version, he can contact technical support if he wants, and they will explain how to set it up; luckily, Twaryna knows English well).
2. The second method is to set up an RTMP Server on one computer.
More details here.
3. Route the traffic through Twitch (if the internet connection can handle it)