Microsoft Edge team posted some details about upcoming changes in Web Notifications in the Edge browser. Microsoft Edge web notifications are available with Windows Insider Preview build 14342. This feature will be available to all users with Windows 10 Anniversary Update. Web notifications keep users informed of new messages by site and plugins. Web notifications are full integrated with Windows 10 Action center.

users can manage permissions at any time under the Advanced Settings menu

Microsoft Edge team posted:

We continue to prototype support for Service Workers and Push API in the EdgeHTML platform, we’re investigating the potential for notifications when the browser is not active or the origin webpage is not loaded. This will also enable notification support on Windows 10 Mobile devices and enable persistent notifications in the Action Center, so that when clicked, the corresponding Service Worker can be invoked to take action.