multiple Chrome instances with multiple profiles
I didn't originally specify, but the idea is you have one instance of Chrome running on one display, and another running on another, and you are logged into the same Google account from both. In theory, this should give you the exact functionality you want. You should be able to close all the windows of one of the instances, and hide the application, and still see the notifications pop up (if the notifications steal focus).
Open up /Applications/Utilities/Terminal.app
type at the prompt
josh2note@stack:~$
josh2note@stack:~$
josh2note@stack:~$ mkdir Chrome\ Profile\ Secondary
josh2note@stack:~$ /Applications/Google\ Chrome.app/Contents/MacOS/Google\ Chrome --user-data-dir=Chrome\ Profile\ Secondary &
josh2note@stack:~$
josh2note@stack:~$
Drag the Chrome window that appears to your secondary display. Log into your Google account.
Go back to the terminal.
type at the prompt
josh2note@stack:~$
josh2note@stack:~$
josh2note@stack:~$
josh2note@stack:~$ mkdir Chrome\ Profile\ Main
josh2note@stack:~$ /Applications/Google\ Chrome.app/Contents/MacOS/Google\ Chrome --user-data-dir=Chrome\ Profile\ Main &
josh2note@stack:~$
Leave the Chrome window that appears on the main display and log into the same Google account.