How to obtain a feed of comments entered through the 'chat' box during a YouTube live broadcast?

It is now possible to return chat messages for your own broadcasts using the LiveChatMessages endpoint as part of the YouTube Live Streaming API.

When creating a new liveBroadcast object, a liveChatId String will be returned as part of that liveBroadcast's snippet. Pass your broadcast's chat ID to LiveChatMessages/list endpoint's liveChatId parameter, and id, snippet, and authorDetails into the part parameter.

HTTP GET https://www.googleapis.com/youtube/v3/liveChat/messages?liveChatId={liveChatId}&part=id%2C+snippet%2C+authorDetails&key={YOUR_API_KEY}

This will return an array of liveChatMessage resources. The actual chat message is contained in the textMessageDetails dictionary as the value for the messageText key.

"textMessageDetails": {
  "messageText": string
}

Ibrahim Ulukaya, a member of Google's Developer Relations team focusing on the YouTube API, stated the following on a similar quesiton (How to get chat content of Youtube live event in Java):

The API doesn't have connection to live chat at this moment. We are hoping to get that content incorporated into API soon.

via https://stackoverflow.com/a/26427743/1085891