I mostly care about alerting when one of my Rachio devices goes offline. However they can track and display more.
For Nest they offer:
Real Time Text and Email Alerts About Conditions
Junction sends real time text and email alerts to Nest users whenever conditions such as temperature or humidity move outside of a user’s thresholds. Junction will also send an alert if the Nest unit stops working or loses connectivity. Now Nest users can know the moment their spaces become too hot, cold, dry, humid or if equipment has failed. It’s an effective way to monitor things like HVAC functionality or windows left open.