These days people use Server Sent Events and WebSockets for this: The client requests data from the server (possibly through a "subscription" type of request), and the server keeps sending messages (e.g. chat messages) until the client cancels the request or closes the connection.
Not having looked deeply into Server Push, it sounds ideal for this sort of thing, and then we could do away with both SSE and WebSockets, neither of which are very HTTP-y.