For TV, last I worked on a system like this the clients received data the same way as non-live streams: Http streaming (HLS or Dash), where you fetch playlists and small video files and the player stitches them all together. There's buffering along the pipe, the 30-60s total delay (which you'll notice if you watch sports and chat with someone who has cable and is watching the same thing) is a cumulative thing, so you don't see a 1-min startup delay, you just near-instantly get dropped into something that's already quite a bit behind.
Not sure what Twitch does. The over-the-network video game streaming-console services are obviously completely different from TV land, they couldn't get away with it there; but for TV the expense of better isn't seen worth it.