What is the crowd's opinion of Open Streaming Platform?

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Check if RustDesk is something similar to what you are looking for. It is supposedly a TeamViewer alternative, so framerate probably won't be good enough for enjoyable gameplay.

Then one other way would be to set up OwnCast or OpenStreamingPlatform (basically self-hosted Twitch) and figure out some way to get your keyboard and mouse inputs to the server.

Or, if you are willing to go with proprietary solution, install Steam on your server and set up VPN, and then use Steam In-Home Streaming / Link / Remote Play / whatever that function is called. That will probably give you the best results.

I set up a local rtmp server to share gameplay privately (and outside of the reach of the major streaming services), then stream to that with OBS.

We've just used vlc as a client, but you can also host a web service frontend if you want to make it more convenient for them (but more work for you).

Some streamers set their phone up as a "webcam" with obs: https://www.gamerebel.net/how-to-phone-webcam-obs/

Once you have your phone streaming to OBS, obs can then send the feed to a streaming server and also record. Once you're set up on OBS, you could use something like open streaming platform to stream it with a nice twitch like interface for people.

I know that https://openstreamingplatform.com/ has some plan's of doing OAuth and I'm trying to write in some LDAP functions to it as well.
So it might be something that could interest you in the future.