Stream Screen from one machine and webcam from another
Looks a lot like  share desktop to laptop to stream from laptop  (that guy's goal was lower CPU load on his desktop, but same solution.)

Use the usual streaming setup on your desktop, but with high bitrate.  stream to laptop, decode there.  overlay webcam, encode for normal streaming, send that output to Twitch.tv.

IDK what it takes to send streams to twitch, or to receive streams from desktop streaming software.  Assuming you can get ffmpeg to do it, and read from your webcam, use the overlay filter to combine the streams, and -c:v libx264 -preset medium to encode to h.264.  Not sure what you'd need to do to get ffmpeg's output to go to Twitch.tv.