To work with push notification youll need a way to implement GCM inside DS either raw or using some provider nativeSDK like Parse or Backendless among others, by creating a brad new plugin for that (and share with the rest of us ;) ) , It will be quite difficult.
On the other hand, as Dave said, you can connect to the service itself and request the update.
I have tried firebase before, Its quite powerful, but the JS api theyprovide its , hmm, awful, AT LEAST FOR ME, so what I usually do is to work with the REST API. Nevertheless polling for updates is not a good idea if you have a real-time requirement..
You may want to try Server Sent Events, droidscript under the hood have a webview that is capable of those. In other words, it lets you create a listener for server events, depending on the api you may need authentication/authorization in a token or something. Particle and Twitter have them..
depending on your need you may need to make request as well, I was able also to use Fetch in DS by adding the Fetch file. Its a big feature and you can configure it pretty easily.
Some info about how to use it.
Where to get it ( there are many versions) but I liked this
Hope that helps