Currently, desktop sharing runs as an applet within the browser. Recently, Chrome stopped allowing applets to run within the browser.
Convert the desktop sharing to an external Java application that can be loaded via a Java Network Launch Protocol File (JNLP). This will make it compatible with Chrome and enable faster desktop sharing as the Java application will have more access to the CPU.
Other desirable features
The feature is now in BigBlueButton 1.1-dev. You can try it out on http://demo.bigbluebutton.org/