There are a lot of options when it comes to screen sharing. Wikipedia has a nice table comparing lots of remote desktop software.
One that looks fairly promising is FreeRDP. The source is on github and it seems to be pretty active project.
If you want to learn more, you can search for info about the Remote Desktop Protocol (RDP) which is what is being implemented in many of these programs.
If you are looking for something more web-specific (like Chrome's remote desktop), check out WebRTC. It is what the Chrome team is using to accomplish their screen sharing.