I just set up our live chat box on our website and i tested a couple of times with test leads. The conversations are not saved under the inbox on hubspot or being sent to my email address. I've checked my spam and remove all the possible hubspot emails from our bounce back/spam section. I spoke to a rep over the phone and I was told everything looks good on our end.
Just wanted to confirm if you can provide us more details on how is set up currently, you want to receive a notification of a new chat via email? This article here, explains more about how to configure your conversations notification preferences.