Total loss of control.
The moment you put something offsite, you immediately lose control of it. It's like the proverbial bank safety deposit box. The media makes these things out as secure and protected and they're not. A bank will give you up just as readily as an American Internet company.
At least in the US, you still retain privacy rights to your stuff so long as it remains in your possession. The moment you let it leave your control, all of this "you need a warrant" stuff evaporates.
Plus there's the usual trust issues of letting someone else manage your stuff.