You're in luck, Comment Notifier's FAQ says:
Comments notifier was designed to work independently of the built-in comment notification functionality of WordPress. The Options->Discussion->E-mail me whenever check boxes have no effect on this plugin. If you have the above options checked, the email address listed under the Options->General->E-mail address field will also receive an email as is standard with WordPress. If you choose to leave those options unchecked, the main administrative email address will not be notified and Comments Notifier will still function and send out the appropriate emails to your list.
So you don't have to get all of those "new comment" emails. They do tend to get annoying.
I used this plugin for a while so that my editors could get these emails, they finally asked me to turn it off because they were getting too many emails. IMO, it isn't need as much now that you can easily see when you have comments in the Dashboard.