A new feature is being rolled out (first tenant and then general availability – expected to be completed by the end of the year) which will notify end-user by mobile push notification (same than the one you get when a new email arrived in your inbox) when someone is sharing content with them using SharePoint Online or OneDrive for Business.

This settings is enabled by default and can be managed by SharePoint Online administrators through the SharePoint Online administration center (within the Settings section)

image