Simple implementation of `hook_watchdog` to send messages to external service.

watchdog.php

function hook_watchdog(array $log_entry) {
  $severity_list = array(
    WATCHDOG_EMERGENCY => t('Emergency'),
    WATCHDOG_ALERT => t('Alert'),
    WATCHDOG_CRITICAL => t('Critical'),
    WATCHDOG_ERROR => t('Error'),
    WATCHDOG_WARNING => t('Warning'),
    WATCHDOG_NOTICE => t('Notice'),
    WATCHDOG_INFO => t('Info'),
    WATCHDOG_DEBUG => t('Debug'),
  );

  if (strpos($log_entry['type'], 'debug') !== FALSE) {
    $output = array(
      'site' => $log_entry['type'],
      'message' => t($log_entry['message'], $log_entry['variables']),
      'status' => $severity_list[$log_entry['severity']],
    );
    drupal_http_request('', array(
      'headers' => array(
        'Content-Type' => 'application/json',
        'Accept' => 'application/json'
      ),
      'method' => 'POST',
      'data' => json_encode($output),
    ));
  }
}