Well of course the EU governments will have special rights. You'll still have to obey EU law after all. But at least there's an extra step before American law enforcement can access that data.
So consider, for most people, it's good enough protection. Because we're not worth the extra step.
What's more, EU law says that personal data can't just be shared with companies or governments. No easy backdoors to your data. I'm no lawyer, but I'd expect that any government, US, EU or otherwise, will have to provide a warrant. Final decision will be with a European judge, not an American one. That means quite a bit.
In many countries, the easy backdoors are already baked in on a constitutional national level.
See the NSA pressuring the German government to modify its G-10 law, or FAD "legal guidance" operations in Sweden and the Netherlands [0].
Case in point: Merkel might have complained loudly about the NSA spying on her phone for the PR, but I'm pretty certain she was fully aware that this was completely legal to do for the NSA [1].
For most things, american law enforcement cannot enforce itself in europe and vice versa, hence the relative "protection" from govt. abuses that remote storage offers.
Yep. This scenario was, in some ways, inevitable. It took nations a little while to catch up to the power that separation of data from a person's physical location via the cloud put in the hands of people and corporations, but they're catching up.
At the end of the day, most users in a few years won't have a choice of no country's government being able to access their cloud state; they may have a choice over which country.
So consider, for most people, it's good enough protection. Because we're not worth the extra step.