Unlike Form, File Upload, or Booking, which are one shot submissions, the Live Chat block is a real back and forth conversation between a visitor and you.
How a visitor's side works
A visitor doesn't need an account. Starting a conversation hands their browser a private key that keeps that one conversation theirs and readable only by them, stored in that browser only, so starting a new conversation from a different browser, or after clearing site data, starts a fresh one.
Replying, as the project owner
Open your project's Inbox panel (inside the editor, using your own CodeBlock Studio sign in) to see every conversation and reply. While you and a visitor both have the conversation open, replies appear instantly, but nothing is lost if one side isn't watching at that exact moment, reopening the conversation always shows the full history.
Tip: Chat messages are personal data your visitors are trusting you with, the same responsibility described in "Collecting form submissions, files, and bookings from your visitors" applies here too.