Nope, you're thinking of this backwards. The people who actually have something illegal to hide might actually try to do what you're saying here, make it untrac

Nope, you're thinking of this backwards. The people who actually have something illegal to hide might actually try to do what you're saying here, make it untraceable somehow (at non-trivial risk and expense, of course!). But if what you're worried about is more casual, warrant-less invasion of privacy by your government (that isn't motivated by something as blatantly illegal as the examples you chose), "just store it at home" is a no-brainer.