Video chat for the home network - school?

Is there a possibility / program that enables a video conference in the local network.

The reason for the question is the current situation in the upper school, in which the teachers are allowed to teach in two separate rooms (classroom teaching). The BigBlueButton platform is used. The picture and audio quality leaves a lot to be desired due to the poor bandwidth…

I thought to myself, it's really nonsense to run the video conference over the Internet. I mean, there's no way to connect the whole thing over the LAN in the school's private network. Unfortunately, the Windows-integrated remote desktop is not a real solution either, since the teachers of the laptop do not have a professional version. In addition, the school PCs are very limited in their settings.

Unfortunately, I can't find anything on the net about how to set up a video switch locally,

Does anyone have any idea?

Thanks in advance,

Desperate Q12ler sends his regards

This also works with X2Go. This is free software. Available for Windows, Mac and Linux. https://wiki.x2go.org/doku.php

Works over the Internet and local home network. Every participant has to install this.

I understand, sounds pretty good at first, but it doesn't really work as a video chat… Does it? I would like to transmit video and sound alternately. Just like a classic video conference.

Would OBS be used? I don't know my way around that well