Sending WordPress database information to cross domain the safe way?

If you have to do this with php then you are going to want to use the WP_Http class. Specifically, you’ll probably want to use wp_remote_post(). Something like this should get you started.

function pushData($data){ //data is an associative array of the things you want to send
    $url = "https://yourhost.com/your_catch_script.php";
    return wp_remote_post($url, array(
        'method' => 'POST',
        'timeout' => 45,
        'redirection' => 5,
        'httpversion' => '1.0',
        'blocking' => true,
        'headers' => array(),
        'body' => $data,
        'cookies' => array()
      )        
}

You will probably want to do some sort of validation on $data to make sure that it is an array and contains values appropriate to be written to the database.

Depending on exactly what your needs are, you may be better off just doing some kind of database replication, you can find a good explanation of how to do that here.