Join.me doesn't use a server, to my knowledge, but instead operates more or less like Teamviewer. People connect directly to your computer, which is why the stream gets slower the more people are in it.

I don't think it'd be too difficult to make something like that. It's - I think? - relatively simple technology. It's just a matter of programming it.

Any idea how?

I'm afraid not. I'm no programmer, or I'd actually have a job right now.

There's also sadly the added problem that programmers don't come cheap. You'd likely be shelling out a lot of money to get this program made if you do pay for one.
But you never know, there might be someone out there willing to work pro-bono.

No, I'll pay. Or at least ask for a serious a quote.

Try www.teamviewer.com

^This. Teamviewer is really popular software in commercial use.

Just found out about it too. Can't wait to try it for my next stream