Can we implement our own Web Push Service?

I am interested in building an entirely decentralied network, with thousands of “validators” listening for transactions being posted on their respective data “streams”. Each validator would run on a regular desktop client (behind a firewall, so can’t listen on a port) and would subscribe to push notifications on a couple computers out on the internet (routing via a Kademlia DHT).

My question is, does Web Push as implemented in Chrome and Firefox allow us to run our own Push Notification service? If not, I can use WebSockets or COMET for Push, but it would be even cooler if I had a service worker loaded even when the website isn’t open. Is it possible to do, or we MUST rely on the browser vendor’s centralized service to deliver the push notifications?

https://blog.sessionstack.com/how-javascript-works-the-mechanics-of-web-push-notifications-290176c5c55d