The browser builtin network monitor only lists all network connections made so far. With push/websockets, the network connection is established once and then reused for all future data transfers. You thus indeed won't see new network connections appearing at all. You should however still be able to see the transferred data in the "Frames" tab of the opened websocket connection. In my case, Chrome has indeed the bug that it doesn't immediately refresh the "Frames" tab on newly incoming data, but you'll see it when you switch forth and back to another tab, e.g. to "Headers" or "Cookies" and then back to "Frames" tab. See also this video.
If you want to monitor all HTTP network traffic, regardless of the browser used, you'd better use a browser-independent tool like Fiddler.