Update the chat example to publish messages on the websocket connection

I think the chat example would be a lot more valuable if the messages were published via the websocket connection instead of as separate HTTP request. I'm trying to migrate from gorilla/websockets but having a hard time figuring out how to both read from and write to a client from different threads. If not publishing the messages themselves, maybe a "typing" indicator when a user is typing in the input? Anything that demonstrates concurrent full duplex communication over websockets really.

Start by locating and running the chat example, then inspect how it currently sends messages over HTTP. The issue proposes moving message publication onto the WebSocket connection, or adding a typing indicator, to demonstrate concurrent full-duplex communication. Confirm which behavior is accepted before defining done.

Written by the indexing model from the issue text.