Comment notify lets your site visitors – signed-in and anonymous – ask to receive e-mail notifications about subsequent comments while they are leaving a comment themselves.

This module has been suggested to and used by Dries, and is being considered for inclusion in Drupal core. That's always a good sign.

Download and install:

Command line steps
drush dl comment_notify
drush en comment_notify

Comment notify requires the core Token module, which, if it has not been enabled, we will be prompted to enable.

Picture: Drupal prompting to enable the Token module, a dependency of the Comment notify module.

And it starts working immediately. There is a settings page where the node types

Administer » Configuration » People » Comment notify
admin/config/people/comment_notify