I agree that people have a legitimate, reasonable right to have their private lives kept private.
I don't agree that public information on the internet that is indexed by Google constitutes private information.
I can see a situation where someone illegally put your private information on the internet and you send a C&D and then get a court to order that website to remove that information and they comply and THEN you ask Google to remove it from search results (assuming it doesn't automatically get removed the next time the index is updated.)
Maybe the website is in a different country and doesn't comply and you want Google to take it down.
Then maybe I could understand an argument for a process to remove private information from Google.
But if you post naked pictures of yourself on a forum or advocate cannibalism on twitter then tough luck. That's no longer private information as you just published it to the world.
It's not like removing the information from their index without removing it from an actual website is going to make the information 'private' again.