Answer
If a customer closes the chat on your website and your team or HelperHat replies, the chat button shows a small red badge with the number of new replies. It shows 1 to 9, and + for 10 or more. The badge appears on every page of your website that has the chat button, including after the customer reloads a page or comes back from the chat page on a phone. Opening the chat clears it.
If your website is open in a background browser tab, its tab title changes to (1) New reply or (2) New replies, with the real number. Your website's own title comes back as soon as the customer returns to the tab or opens the chat.
While the badge shows, the X that hides the chat button is hidden too. Screen readers hear about new replies once, and the chat button's name includes the count, for example Open HelperHat chat, 2 new replies.
The website checks for replies about every 20 seconds, and less often in a background tab. It only checks for customers who sent a message in the last day, or the last three days while they wait for a person, in the same browser they chatted in. It receives only the number of replies, never the message text, and notes that only your team can see are never counted. HelperHat doesn't email customers about new replies.