There isn’t any built-in function for DMs. You’d need to integrate something like AtomChat or FastComments.
It’s possible to build email notifications with automations in Airtable. For example, it wasn’t hard to build some scripts that send out instant notifications and nightly digest notifications for comments.
If I needed this, I’d probably try FastComments first. You can see it integrated into Softr on unlock-softr.com. I helped Joachim a bit with the integration, and it seemed like a good commenting tool with DM functionality.