I just tested my store's blog out myself, and nope, customer's do not get notifications when you write a new comment. At this point, I'm starting to wonder if allowing comments is even worth it. I can't expect potential customers to come back and check every couple of hours to see if I've replied yet. Instead, they'll likely just think that I'm unresponsive, since they never got an email notifying them of my response - or anyone's response for that matter.
I'm chiming in too to add responding to comments as a basic function of the blog (without having to install another app). I like the clean look of the included commenting function, but it would be great to get an email notifying you when a new comment is left and then be able to respond directly to them so it appears under their original comment. I just switched my blog over to my Shopify site (years of posts) and never thought I wouldn't be able to respond to comments! A heads up that this isn't an included function would be great - especially if Shopify isn't planning on offering this functionality (seems this has been a feature request since at least 2014 based on the comments I've gone through). To be honest, this is quite discouraging.