For each new Conversations we are getting email notification, my suggestion is the email notification just show the url and title, it will be better if the conversation text also included in the email. As this was there on the old Private Message.
You can disable the email notices for conversations here if you want: https://forums.digitalpoint.com/account/contact-details As far as putting the message in the email, there are some technical issues with that currently (see this post).