There is an Alexa skill (not developed by me) now called Notify Me which uses the preview Alexa push notifications API. This particular skill has a backend RESTful API, that you can post to in order to send notifications to your own devices. You enable the skill and link your account and you are sent an access code to use the API. I have got this configured now, using the rest notify platform, and it works well:
- name: notify
The only disadvantage, which is a limitation of Alexa push notifications, is that when you get a notification the Echo will just ding and light up green. You need to say something like “Alexa, what are my notifications?” or “Alexa, what did I miss?” to play the notifications.