I think one solution that deserves a usability study would be a "designated notification queue widget." This would alleviate some of the concern that users will suspect malware activity because it will clearly be an integral part of the desktop. It will be a stack or queue that remembers the last 'n' notifications (complete with dialogs, if possible) and allows the user to scroll back through any that were missed. It would be minimal and unobtrusive by integrating into the taskbar or desktop and could change color to indicate new notifications since the last time the user checked the widget. And best of all, it could contain an obvious toggle button for "Do Not Disturb" mode!
I do not know if anyone else has ever proposed this, but feel free to take this idea and run with it.