I may have an answer for you...
I actually built a plugin that does just this, but it is for self-hosted WordPress installs.
However, I found that it may be possible to make it available for WordPress.com users if you request it. Check out the Plugin Requests section at the bottom of this WordPress.com Plugin Support page.
I'll give you an overview of the plugin though in hopes that it can be added. It is called Roost for Bloggers.
This plugin uses Roost to send notifications. What is really cool about the plugin is, anyone that visits your site on Safari in OS X Mavericks will receive the prompt for browser notifications, but you can also allow other visitors to sign up and receive push notifications to their mobile devices. The plugin also has an auto-send feature that sends notifications when you post new content.
The plugin and service is free for bloggers. No catch. Check out the Roost for Bloggers plugin in the WordPress.org directory. Also have a look at the Roost site: http://roost.me.
(Full-Disclosure: I work for Roost.)
I will answer any questions here, or via email. My email is dan.stever at roost.me.
As a teaser: we are actively working and updating the service to support other push-enabled browsers. Chrome should be available fairly soon, but I do not have a solid date.