In a one PC, multi-display arrangement, you have the ability to specify in which display Composer's Play Mode will first run.
NOTE: To cycle through each possible display one at a time, use the F11 key on a physical keyboard.
The trick is to launch Composer using its desktop shortcut. Drag that shortcut to the display in which you want Play Mode to occur. Then launch Composer by double-clicking on the shortcut. Play Mode will use that display as the primary display even if Composer is subsequently moved to a different display.
For example, double-clicking on the Composer shortcut while it's visible in Screen One will launch Composer and cause Play Mode to use Screen One. Double-clicking on the Composer shortcut while it's visible in Screen Two will launch Composer and cause Play Mode to use Screen Two. And so on.
In fact, you can use this method to run multiple instances of Composer at the same time with each Composer's Play Mode assigned to a different display. Simply drag Composer's shortcut into Screen One and double-click, then drag into Screen Two and double-click, etc.