A storage locker or bank deposit box is private property even if it's effectively held by a 3rd party.

There's no reason the same doctrine couldn't extend to digital papers and effects.

I'm not a lawyer or anything but the third party doctrine holds because you're willingly and knowingly giving your information to said third party. I expect that the difference there is that the contents of a storage locker or safety deposit box are explicitly confidential from the company their are leased/rented from.