I have 2 LCDs, but game on only 1 (the left side). The second (right side) just sits there until I exit game. Shows desktop and anything there (handy if I need to leave a picture of a map open for events etc).
However, when testing 2.19 offline just now I noticed something. I was moving my mouse to the lower right corner to take a screenshot offline and noticed not only was my cursor in AH moving, but there was a second desktop cursor matching its movements on my right monitor! It wasn't too far into the monitor, maybe 200 pixels or so at most, but with my AH cursor still in play I clicked in-game and it acted as if I was clicking my desktop one.
It minimized AH and dropped me to the desktop! AH resumed normally when I clicked the task bar for it, but that was rather odd.
Hope that gets fixed.
EDIT: Oh, a couple of other details. Both monitors' native resolution is 1280x1024, and both are set to that. AH is set to use 1280x1024 so that it takes up exactly 1 monitor.