May I redefine you question as below
Can we have our own back-end to send push notification to Chrome, Firefox, Opera & Safari?
Yes. By today (2017/05), you can use same client and server side implementation to handle Chrome, Firefox and Opera (no Safari). Because they have implemented web push notifications in a same way. That is Push API protocol by W3C. But Safari have their own old architecture. So we have to maintain Safari separately.
Refer browser-push repo for guide lines to implement web push notification for your web-app with your own back-end. It explains with examples how you can add web push notification support for your web application without any third party services.