I’ve fixed this by adding:
echo '<div class="mgm_message_error">' . $errors['error'][$code] . '</div>';
instead:
$error_string .= '<div class="mgm_message_error">' . apply_filters('login_errors', implode(' ',$errors['error'])) . "</div>\n";
Now, error messages and success messages are displayed, the only problem is on register page. Plugin prints same error message twice, and I still can’t figure out where is the problem.