Private domain registration has nothing to do with Google. It just means no one can look up the domain in a whois search to find out who owns it along with personal information such as your address. This is usually a good idea to protect your privacy and help guard against identity theft.
In your Dashboard under Settings > Privacy you can decide whether or not to allow Google and other search engines to index your site. Currently you have it set to allow search engines. That should accomplish what you are wanting to do.