Solution 1:

There are several software that help you doing that. For example:

  1. http://www.ws4gl.org/ Webcam Studio for stream your video from different sources, also from desktop.
  2. http://recordmydesktop.sourceforge.net/about.php
  3. https://launchpad.net/kazam It seems pretty good.
  4. http://xvidcap.sourceforge.net/ I used that when i needed to make a video of my desktop

And for another list of available software check that post: http://ostatic.com/blog/six-top-screen-capture-tools-for-linux#intro

Hope that this help.

Solution 2:

VLC should, in theory be able to take a video stream from your desktop- with "screen://" as your input option and output it to another streaming source. However whether if will work with the software you have in mind would depend on what that software is.