How do I run an app in Full Screen mode on OS X Lion on my second monitor?
This is an excerpt of Apple's AppKit Release Note for Lion regarding full screen apps &amp; multiple desktops.


  Multiple Monitors and Full Screen (New since early 2011 seed)
  
  We have made some simplifying assumptions for Lion Full Screen
  behavior on multiple monitors. Multiple monitors are treated as a
  single unit by Spaces, and therefore are also treated as a single unit
  in Full Screen. This means that all monitors will be dedicated to
  windows belonging to the full screen application, and there can be
  only one primary full screen window visible at a time. A secondary
  monitor is useful for inspector windows.
  
  Secondly, because the menu bar is located on the main monitor, the
  primary full screen window will be located on the main monitor as
  well. This allows the menu bar, floating toolbar, and full screen
  window to maintain their interrelationship on the same monitor.


The full document can be found here:
https://developer.apple.com/library/content/releasenotes/AppKit/RN-AppKitOlderNotes/index.html#10_7FullScreen

According to this document, your full screen apps will always be shown on your primary monitor, and there is nothing that can be done about it.