The phone doesn't poll, instead it goes to sleep, and gets awakened by the push notifications. Just like GCM/FCM ones. Part of the key thing, because not any one can self host, is that it requires very little information in comparison, and some providers are open source and even free SW. The one I use is ntfy, there's the next cloud (next push), and not long ago you can use the conversation push service (up.conversation.im) through the Convesations xmpp client. I was aware of Conversations capable of becoming a unified push distributor, but actually was looking for it to use a unified push distributor instead.

But I was informed already it's not necessary, and doesn't make much sense, by being very low power consumer, even though it requires to keep unrestricted battery consumption on the background. So no issues by Conversations not supporting using a unified push notifications distributor.