Part of this behaviour depends on an activation AGENEON completes before it goes live for your workspace. The page in your workspace states the current state truthfully.
Four states
A conversation is with the Agent, waiting for a person, with a person, or back with the Agent after a person handed it back. The Agent asks for a person when your operating settings say so or when a customer asks; the Inbox shows every conversation waiting for a person first.
Replying as a person
Take a conversation and reply from the Inbox. While a person holds it the Agent stays silent. Hand it back when you are done and the Agent continues with the context of what you said. Every message shows whether the Agent or a person wrote it. Website chat is the exception: a person cannot reply inside the website chat, so reach a website customer on the phone number or other contact they shared in the conversation.
Delivery
A person's reply on a provider channel is delivered through that provider and depends on the channel's activation. A send whose outcome the provider did not confirm is shown as unknown rather than as sent, so you are never told a customer received something that may not have arrived.
Verified against the product at PLC-7