This workflow is possible using an external synching service, which will basically sync any edited file within a particular folder. I am right now using one such service, which syncs my whole vault folder on my Desktop and Phone. So the data will be always in sync, on multiple devices.
Now, coming to the feature of getting push notification. The external service you are using for this functionality, lets consider Obsi app here. The Obsi app will be required to run in the background all the time, so that it scans as soon as any file is updated by the synching service. This is possible on Android, but almost impossible on iPhone.
So, technically, for actual real-time collaboration either Obsidian should be running on both the devices all the time, to get real-time push notifications. And using plugins like Relay, it will be possible. Otherwise the only option remains is using a server, which will send the push notification to phones even if Obsidian or any other app is not running in the background. (I think relay plugin is also using a server.)