Irene, the email Debug Mode is exactly that.
It refers to the emails requested by people using Notify me when a reply is posted, in the posting panel, and Notify me about PMs, in their profiles.
When they get the Debug Mode message it means that the emails are not sent, but the messages are posted, and the PMs come through.
The worst thing is that the same thing happens when someone registers. They will get the Debug Mode message, and their Activation email will not be sent.
So, the best thing to do is to put some info about it on the index page, with an email where they can reach you, so you can activate their accounts manually. (That kind of info is always good to have).
The problem as such is reported to the Webmaster already, as it is on a few domains. I'll add .net to the list.
FoxLeonard