How does communication work in Chrome Remote Desktop?

Question

Google

I want to write a "small" application for Chrome and Android with the maximum use of Google resources.
Applications on the phone and on the computer must somehow synchronize with each other to store a common database. To implement this task, I was attracted by the work of the Chrome Remote Desktop (Chrome Remote Desktop), but here's how it works and how to implement the same on your application?
To be clear:
There are Application_PC1, Application_PC2 and Application_TEL that store their addresses on the Google server associated with the user's account. If there is a need for any application to exchange information with other user applications, then it requests addresses from the server, contacts the applications and performs the task.
For example: The phone (Application_TEL) was in the off state while charging, on a stationary computer (Application_PC1) the wife plays solitaire, and the user on a laptop (Application_PC2) makes changes to the database.
As a result, Application_PC2 finds the application on the personal computer through the Google server and reports changes to the database, Application_PC1 makes adjustments to its database. After turning on the phone Applications_TEL through the Google server finds other applications and checks for changes in the database, if any, then changes are made.