Chat transcripts are not sent automatically, as some people do not prefer unnecessary spam in their Inbox
However, there is an option for you to receive the chat transcripts if you wish, the steps are :
1) Click on the 3 Dots at the top left side corner of the current chat window and click on "Send transcript"
2) Enter your email address and click send
3) Chat transcript will be sent to the email address entered by you, once the chat is finished