I see what you mean now. This seems to be a limitation of how full screen mode is implemented for browsers, as the same thing happens with Safari as with Firefox -- as soon as I click anywhere on screen 1, the display on screen 2 reverts to view-in-window mode.
I have no workaround for YouTube but for Hulu, you might want to try
Hulu Desktop, a stand-alone application that in some ways is much better than viewing Hulu content in a browser. Just make sure your Mac meets its system requirements before you try it to avoid problems.
It is my preferred way to view Hulu stuff. It works perfectly in full screen mode on a second monitor -- as I type this on screen 1, I'm watching an old Avengers episode fullscreen on screen 2 -- although you may get a notice the first time you drag its application window into screen 2 that a resolution change has occurred & you should quit & restart it. Do that if needed & then just tap command + F for fullscreen mode. When you click back on screen 1, it will not drop out of full screen mode.
In fact, if you are running 10.6.2 & have an Apple Remote, you can use it to control Hulu Desktop without having to make the app frontmost to use the keyboard or mouse to control it. How cool is that? 🙂