Most libre applications implement their own websockets for this. EDIT: some propriety spyware like WhatsApp use those too, can confirm myself, as I unfortunately have to use it on my GrapheneOS owner profile (which ofc doesn't have the sandboxed GPlay services installed; I could technically use it on my profile for personal proprietary apps, but it would be a major pita) ENDEDIT
(unfortunately Proton uses only Google FCM in most of their apps and doesn't publish them on F-Droid or Accrescent)

However, those can take a strain on your battery since every websocket maintains it's own connection.
This is where UnifiedPush in combination with either nfty or Sunup comes in
(EDIT: I use ntfy myself, also Sunup seems to be quite new, so it might come with some quirks; EDITEND there is also a plugin for your own Nextcloud)

Since Unifiedpush maintains only one connection for every app using it, this takes quite a bit less charge to upkeep
(it's explained quite well on their website, which is hyperlinked)

If you are looking for Unifiedpush support on Signal: Molly is a hardened fork of the client and does support it through Mollysocket

EDIT: If you use Telegram there is Mercurygram. As for Matrix clients, I use Element and Schildichat. All these support UnifiedPush