Always give users control
Our inboxes are overloaded as it is. You're right to worry. You have to provide notification for those who come and go, but you can't force on the regulars or people who end up linked to a popular discussion.
Stack Exchange provides a good pattern
A checkbox to activate email subscription:
And in-line settings to control frequency and delivery address:
How it applies to chat
A chat or real-time discussion app can follow the same logic, but it should exist in an easily discoverable (or directed) settings panel/page. The user needs to have notification options for
- No notifications: this should be the target if you aspire to a "sticky" experience
- Time period "digests": hourly, daily, weekly.
Don't forget the user who loathes email. You could conceivably support SMS, direct Twitter messages, Snapchat, or some other messaging service of the moment.
The Twitter model
I think it's over the top, but Twitter is a good conceptual match for what you're doing and supports extensive email notification tweaking. This is all real time updating, but the idea of event-based notifications might fit your app.
Fitting the application
In response to your follow ups ...
It's rare to find a model in the wild that perfectly suits your unique intersection of use cases (unless you're copying someone entirely). You'll always find exceptions to the other app's rule.
In your app, it sounds like you'd want an option for
Notify me of the first message from a user
Notify me of the first message from a user in ___ days
I'd also challenge the assumption you mentioned that users won't want to limit frequency to digests. If your site generates a lot of activity (who doesn't want success?) users may only want to get sidetracked by a new chat at certain intervals, like I mentioned in point 2 above. (Time blocking is a well-known practice among people more efficient than me.)