Google Home may be getting notification support in the near future, at least if the recent appearance of a notification toggle in the Google Home app is any indication. The option appeared under the Google Assistant section of the Google Home app's settings for some users and the text accompanying the option says that users who turn it on can expect to receive notifications for things like weather, and that even if the setting is turned off, users will still hear alarms and important alerts. The text seems to imply that the notification toggle is only meant to be compatible with contextual notifications that you would normally receive from the Google App on your phone, but it is entirely possible that the feature will also support notifications linked to your Google account like the ones sent by Gmail, Hangouts, and Chrome.
The notification option at hand is only popping up for some users, but those who did receive it have not reported anything changes if they activate it, which could mean that it's just dormant for now and Google plans to activate it soon, or the option went live too early and will soon be removed until the actual feature is ready to be released. Right now, Google Home does not provide any sort of notifications, though users can opt to receive notifications about Google Home via email. Unless you're willing to perform some serious tweaking through Tasker and other apps, the system doesn't speak unless spoken to.
Regardless, those who want to receive notifications on their Google Home right now can do so by using Tasker and an app called AutoCast, along with some community-made plugins. A combination of those allows you to set up your phone to automatically cast notifications that you choose to your Google Home. By setting up profiles in Tasker and hooking them into AutoCast, users can pick exactly what types of notifications they want to receive through their Google Home, and what apps they want to be able to hand those notifications over to Tasker to be put out through AutoCast. Of course, this hack only works as long as your phone and Google Home are on the same Wi-Fi network.