I've read from the Signal team before (no reference links on hand, sorry) that Signal servers send an empty push notification to your device to wake it up, at which point the Signal app fetches your notifications/messages from Signal's servers.
At least, that's what I read for Android/GCM, I'm not sure if that applies to Apple.
I believe they’re referred as “Silent notifications” on iOS. It’s very convenient to update state in the app without having to wait for the user to perform the action manually.
Which would be consistent with most messaging apps that use notifications. Although I'm sure there is an exception to the rule most only use notifications to indicate to the client that a new message is available on the server.
At least, that's what I read for Android/GCM, I'm not sure if that applies to Apple.