This is a problem Android, which is also Linux-based, has already solved. You can wake up a process (on a power efficient core, if available) without waking up the screen.

One problematic issue I can see the Pinephone have in regards to this is that there's no unified push service that works for all apps. Google has forced everyone on Android to use their push service and it's terribly restrictive for application development but over the years this has also solved a lot of battery life issues. Apps holding open connections of their own just to see if there are any notifications is fine on desktops but it's a challenge open platforms will need to solve somehow.