User’s total comment count excluding comments on their own posts

I’m not aware of any WordPress functions that do this, so you can try to play with this kind of query (untested):

function get_custom_user_comments_count( $uid ){
    global $wpdb;

    $sql = "SELECT COUNT(*) as total 
            FROM {$wpdb->comments} as c  
            JOIN {$wpdb->posts} as p ON p.ID = c.comment_post_ID 
            WHERE c.comment_approved = '1' 
            AND p.post_status="publish"  
            AND p.post_type="post"  
            AND p.post_author != c.user_id            
            AND c.user_id = %d";

    $comment_count = $wpdb->get_var( $wpdb->prepare( $sql, $uid ) );
    return $comment_count;
}

and use it like this:

global $current_user;
get_currentuserinfo();
echo 'Total Comments: ' . get_custom_user_comments_count( $current_user->ID );

This assumes that the comments are written when the users are logged in, so the user_id column in the wp_comments table is populated.

Update:

Another approach would be to collect the comments count per user in the user meta.
You could try for example to hook into the comment_post action. This approach could be implemented before the users write comments on the site.