The conversation user is stored on the conversation object in the bot's database as the relationship #speaker.
If the user is not connected this would normally be stored as anonymous, but you can also pass the user name and optional email address through the web API.
The chat API takes a message object that has an info variable that you can set.
var request = new XMLHttpRequest();
request.open('POST', 'https://www.botlibre.com/rest/api/chat', true);
var xml = "<chat application='12345' instance='12345' info='John Doe [email protected]'><message>hello world</message></chat>";
It will show in the chat log as "info: user email"