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
}