The best open source alternative to Apple Screen Sharing is Remmina. If that doesn't suit you, our users have ranked more than 50 alternatives to Apple Screen Sharing and 17 is open source so hopefully you can find a suitable replacement. Other interesting open source alternatives to Apple Screen Sharing are DWService (Free, Open Source), mRemoteNG (Free, Open Source), UltraVNC (Free, Open Source) and TightVNC (Free, Open Source).
Remmina is a remote desktop client written in GTK+. Remmina primarily targets system administrators and travelers that need to work with computers remotely in front of either large monitors or tiny netbooks.
DWService is an open source project which offers a service that allows access to remote systems (Windows, Mac, Linux, Raspberry...) using a standard web browser - without any component required on the client side.
UltraVNC is easy to use and free software that can display the screen of another computer (via the Internet or a local network) on your own screen. The program allows you to use your mouse and keyboard to control the other computer remotely.
TightVNC is a free remote control software package. With TightVNC, you can see the desktop of a remote machine and control it with your local mouse and keyboard, just like you would do it sitting in the front of that computer.
TigerVNC is an advanced VNC implementation. It is based on the fourth generation of VNC. TigerVNC also includes features from the TightVNC and TurboVNC projects. This includes accelerated JPEG compression. TigerVNC supports the latest X.Org X server.
Dayon! is a user friendly, open-source, cross-platform solution, that allows controlling of remote computers. It can be used as free alternative to various commercial remote desktop and remote assistance products.