Google is now testing a new feature in the Google My Business area that lets customers initiate a chat with your business from your local listing in Google search or Google Maps. This was first spotted by Dino Basaldella who posted it on Google+.
Here is a screen shot of the local knowledge panel with the "message" button and icon:
It says on the bottom the typical response time to get a message back from the business. In this case, it says 5 minutes, which seems very poor.
There is a help document that explains this is a pilot program for "a select group of business owners." You will be notified via email about this pilot, if Google chooses to contact you. It says:
Start chatting by clicking the link in the pilot invitation email from Google. Once you click through, you’ll see a step-by-step guide on setting up your Google My Business account for real-time messaging.
Make sure that you provide a phone number that can receive SMS messages and will be readily accessible for responding to customers.
Google powers this via SMS or their Allo app. I guess that is one way to get people to use Allo?
Google does offer messaging via AdWords ads - so maybe this is just an extension on that?