Hide comments awaiting moderation from user who submitted the comments

Check in your theme’s comments.php for the wp_list_comments() function. In the Twenty Eleven theme, for example, uses a custom callback function which is in the functions.php file and outputs the template for comments.

Within this comments template, you can then use wp_get_comment_status() and only display a comment if it was approved. Example from the Codex:

$status = wp_get_comment_status( $comment_id );

if ( $status == "approved" ) {
    // the rest of the comment loop
}