My goal is to send Gmail-style desktop notifications in Chrome or Firefox from a web app (let\'s call it X) that is NOT currently open in the browser, without requiring the
You can send push notifications even when your web page is not active using Service Workers, the Notification API for service workers and the push API for server-initiated notifications (or scheduled notifications).
As of June 2016, Service Workers are supported in Chrome, Firefox and Opera. See the status at https://jakearchibald.github.io/isserviceworkerready/
See the following links for related discussions, status of implementation and specifications.
You can do exactly what you are looking for using the W3C Push API.
If you want to build everything from scratch I suggest to start reading this tutorial by Google. It is for Chrome, but Firefox works in a very similar way.
However it's a lot of work and the "standard" is still evolving: I suggest that you use a service like Pushpad (I am the founder).