I have a MBA connecting via HDMI to Samsung Monitor. Problem is desktop background for Samsung monitor is not showing. There is only grey desktop and doesn't have any changes even I try to change the image of the desktop at System Preferences. Changes for the MBA desktop is ok. Is that a mountain lion bug?
OS X allows you to select different images for each display. When you open the Desktop section in Desktop & Screen Saver Preferences, you should get one dialog on each screen for choosing a desktop image. Each dialog controls only the image for that screen, so make sure you're setting the image you want on both screens.