I have setup Push Notifications for Chrome on Android. Push notifications go out well for messages and many notifications. However, there are some issues I have seen;

* No push notification when a Friend request is received. However, when Friend request is accepted, the sender gets notification

* No push notification for invitation to an event or a group

* Also no notification when invitation to event accepted (maybe this is not a feature in the software)

Please let me know if these issues are due to configuration, bugs, or as designed. I really would like the Friend request notification, which you mention should work. Also, I would like a notification for request to an event or join a group.