We have a requirement of creating Push Notifications in one of our SharePoint Portals. Expecting the functionality same as in Facebook. We have many communities, blogs and News feed and many more.
You might want to take a look at some existing solutions for push notifications before heading into creating own creations. For example the Newsfeed Notifier from the Microsoft's AppSource is a commercial app for O365 & SP2013+ products, which notifies the users about Newsfeed updates. The app is free for max 5 users and costs monthly beyond 5 users. Note that I've not tried this app nor have anything to do with it.
If you wish to start from scratch, have a look at SP.UI.Status.addStatus-method (documentation for SP2010 but the method is available in newer versions too) With SP.UI.Status.addStatus can be added status messages to the page. From where the messages are received is something you'd have to implement your own logic around. SP.UI.Status.addStatus has some related questions here and here.