We have a Lightning Canvas App component that's an iframe into a web page that normally pushs notifications to the browser. When in the Lightning Page context, those browser notification don't reach the actual browser, and therefore we need to communicate these from the iframe to the encompassing Lightning Page.
As per an earlier post, I haven't found a normal way to communicate information from the Canvas App iframe to the parent Lightning Component.
Update May 2020
You can send messages via a WebSocket, which can now be whitelisted as a CSP Trusted Site in Salesforce, which sounds like we'll be able to communicate to the Aura component to prompt a browser notification.
With this in mind, the real question is: How do I create a browser notification from within a Ligthning Aura Component?