I don't think you're going to be able to do this. If MAME loses focus it doesn't accept inputs, IIRC. Hopefully someone will chime in if i'm wrong.
The closest you might be able to do is to get the menu to load on one screen, and a game on another. You would do this with by altering ini files to target different displays. Errr, i don't have any ini files lying around at work, so i'm going to guess at line names, but... you go into your mame.ini file and change the display1 line from Auto to //display_1// or whatever your display is called in the ArcadeOSD program. You can open that program and look up the top to check. Then you copy the mame.ini file into the ini directory and rename it vertical.ini and change that line to target //display_2//. This will push all vertical games to a different display. You can target things in more detail using horizontal.ini, any machine.ini like neogeo.ini or gamename.ini like SF2.ini.
The problems with this might be that it only works from commandline launching, or the MAME menus might freeze, or simply go black on your first screen when a launched game output is pushed to another screen, or that the menus load up on that other screen if you shut MAME down wrong, which might give you problems with resolution. You're going to have to play around and see what you can get away with.