Rolf Allard van Hagen

This plugin could also have been called Don’t bother the Site Administrator with every other author’s Comment Moderation Notifications, unless the author has no moderation rights but since that’s a slightly longer title, I chose to keep it as simple as I could. However, it explains well what this plugin does:
Normaly, when a comment gets submitted to a particular post, the author of that post gets a notification about it. But when that comment is held for moderation (which depends on your sites comment settings) then the moderation notification is sent to both the post Author (if he/she has moderation rights) and the sites Administrative moderator e-mail address as configured under Settings > General at the same time.
For many blogs or sites where the owner is the only author and his/her account uses the same e-mail address as the Administrative moderator e-mail address, this will boil down to one message to one address. But when the Site Admin is not the only author, like on colaboration sites or sites managed by a webmaster or designer where other people like the client usually posts, this might result in flooding the admins mailbox with moderation messages that are not really his/her concern. The site admin, with enough on his/her mind already, is bothered with each and every new comment in the moderation queue.
This plugin changes that.
Just install and activate it: All post comment moderation notifications will be sent only to each respective Post Author. If, by any chance, the post author has no moderation rights (Contributor level) or there is no author e-mail set then the default site e-mail address will still get the notification.
WordPress Multisite compatible, per-site or network activated or as a must-use plugin.

Comment Moderation E-mail only to Author

Works great

Still works !

Send comment moderation notifications ONLY to the Author, not to the site Administration address any more.

AppJetty

Live comment notification plugin can enable the admin to install a live notification toaster to notify all online customers whenever any comment is made by any online customer.
<ol>
<li>We can have live notification toaster if any user will comment on any content of blog.</li>
<li>Comment moderation must be off to work this plugin well.</li>
<li>If moderation is on then we can have only notification from registered user’s comments.</li>
<li>We can select post type for which we want this notification to work.</li>
<li>We can have multiple comments if comments are posted on multiple posts.</li>
<li>This plugin tracks IP of posted comment and according to it,gives notification.</li>
<li>There is setting in admin,if they want notification open for all users OR only authors.</li>
<li>If they select only authors,then if any comment posted on logged in user’s article,then they will receive notification.</li>
</ol>

Live Comment Notification Toaster

This plugin can enable the admin to install a live notification toaster to notify all online users whenever any comment is made by any online user.

Comment Moderation E-mail only to Author vs Live Comment Notification Toaster