Separate the Challenges As Much As Possible

Speaking generally, you're always going to have the issue of lag when you're, essentially, sharing your desktop. There are lots of solutions out there that aren't even in the game space. TeamViewer, ReadyTalk, BlueJeans, Skype for Business (used to be Lync), Skype, Zoom; and I'm sure there are many more that I've missed or will soon be offered by some startup down the street.

Of those listed above, I believe people have had the most luck with Zoom. If it was me, I'd start there and see whether I could get sustainably small lag on your friend's side. It also has the benefit of scaling when you might eventually want to do audio through it.

The other big challenge is computer and network overhead. FDRpi only mentions recording (intended for YouTube?), implying not streaming, but if one does intend to stream it as well, you're basically doubling the data you're pushing, and to separate sources (your [Twitch] ingest server and the screen sharing provider's server). Results may vary, but multiple computer setups may well make alot of it easier.

Nothing beats real data points, though. As WHiT3R4bBiT says, plan while moving. Start a Zoom trial, test sharing your desktop with your friend while recording to disk with Obs, and see how many kinks you can work out. This weekend. And have fun doing it.

Sidebar: For many things, including some of this audio use case, Discord is doing a better job at it than Skype, so make sure to check that out depending on what you need. Also, if Trumpetmcool is around, he knows this stuff, and I remember some DK who works at a company that does perhaps exactly this thing (for businesses and such). Maybe Eliot Miller? Can't remember, but they are a Bay Area company.