Use comments as notes, visible for comment author only

I used _s as a template. The comments are loaded with wp_list_comments( array( 'callback' => '_s_comment' ) ); and then comments are styled with _s_comment(). That is what you will use.

You can use get_current_user_id() to get the current user id.

$current_user_id = get_current_user_id();

You can get the user id of the commenter with $comment->user_id. You then check that they both match.

if ( $current_user_id == $comment->user_id )

The two changes that I made are in line 11 and 20.

I have added the code in a gist and below.

<?php
if ( ! function_exists( '_s_comment' ) ) :
/**
 * Template for comments and pingbacks.
 *
 * Used as a callback by wp_list_comments() for displaying the comments.
 */
function _s_comment( $comment, $args, $depth ) {
    $GLOBALS['comment'] = $comment;

    $current_user_id = get_current_user_id();

    if ( 'pingback' == $comment->comment_type || 'trackback' == $comment->comment_type ) : ?>

    <li id="comment-<?php comment_ID(); ?>" <?php comment_class(); ?>>
            <div class="comment-body">
                    <?php _e( 'Pingback:', '_s' ); ?> <?php comment_author_link(); ?> <?php edit_comment_link( __( 'Edit', '_s' ), '<span class="edit-link">', '</span>' ); ?>
            </div>

    <?php elseif ( $current_user_id == $comment->user_id ) : ?>

    <li id="comment-<?php comment_ID(); ?>" <?php comment_class( empty( $args['has_children'] ) ? '' : 'parent' ); ?>>
            <article id="div-comment-<?php comment_ID(); ?>" class="comment-body">
                    <footer class="comment-meta">
                            <div class="comment-author vcard">
                                    <?php if ( 0 != $args['avatar_size'] ) { echo get_avatar( $comment, $args['avatar_size'] ); } ?>
                                    <?php printf( __( '%s <span class="says">says:</span>', '_s' ), sprintf( '<cite class="fn">%s</cite>', get_comment_author_link() ) ); ?>
                            </div><!-- .comment-author -->

                            <div class="comment-metadata">
                                    <a href="https://wordpress.stackexchange.com/questions/121354/<?php echo esc_url( get_comment_link( $comment->comment_ID ) ); ?>">
                                            <time datetime="<?php comment_time( 'c' ); ?>">
                                                    <?php printf( _x( '%1$s at %2$s', '1: date, 2: time', '_s' ), get_comment_date(), get_comment_time() ); ?>
                                            </time>
                                    </a>
                                    <?php edit_comment_link( __( 'Edit', '_s' ), '<span class="edit-link">', '</span>' ); ?>
                            </div><!-- .comment-metadata -->

                            <?php if ( '0' == $comment->comment_approved ) : ?>
                            <p class="comment-awaiting-moderation"><?php _e( 'Your comment is awaiting moderation.', '_s' ); ?></p>
                            <?php endif; ?>
                    </footer><!-- .comment-meta -->

                    <div class="comment-content">
                            <?php comment_text(); ?>
                    </div><!-- .comment-content -->

                    <?php
                            comment_reply_link( array_merge( $args, array(
                                    'add_below' => 'div-comment',
                                    'depth'     => $depth,
                                    'max_depth' => $args['max_depth'],
                                    'before'    => '<div class="reply">',
                                    'after'     => '</div>',
                            ) ) );
                    ?>
            </article><!-- .comment-body -->

    <?php
    endif;
}
endif; // ends check for _s_comment()