Is there any open source screen sharing (or desktop sharing) software that I can use to integrate into my web site? Any flash/java based solutions?
Flash does not have public / documented APIs to make this work. Adobe Connect even needs an additional plugin for screen sharing.
I believe Adobe has been working on APIs to help make screen sharing a reality. I believe such features were rolled into LiveCycle Collaboration Service; however I thought that was only available as a hosted service. [And I don't see the screen sharing feature documented on that page]. Here is more info from when the feature was announced; it also confirms that this is part of LCCS.
Beyond that, you can look at some tools like these, which allow you to broadcast your web cam feed. And then tie them into something like Flash Media Server or Red5 to stream that out to someone else.