Social chat system for static HTML sites?

I've been searching for a chat system for my static HTML page. What I have in mind is a user will login to their account (for example, Messenger Chat Plugin), and I will login to mine, and then we can chat directly in the website without having to go to a Business Page, like in the Facebook Plugin.

Everywhere I look, it seems like it's only for businesses, where people can chat your page, and you have to reply to them on their apps (again, like Messenger). I'd like to be able to reply within the site too.

Does anyone know of such feature/widget I can embed? It's not for business, just personal chat system.