  In two-party quantum communication complexity, Alice and Bob receive some
classical inputs and wish to compute some function that depends on both these
inputs, while minimizing the communication. This model has found numerous
applications in many areas of computer science. One question that has received
a lot of attention recently is whether it is possible to perform such protocols
in a private way. We show that defining privacy for quantum protocols is not so
straightforward and it depends on whether we assume that the registers where
Alice and Bob receive their classical inputs are in fact classical registers
(and hence unentangled with the rest of the protocol) or quantum registers (and
hence can be entangled with the rest of the protocol or the environment). We
provide new quantum protocols for the Inner Product function and for Private
Information Retrieval, and show that the privacy assuming classical input
registers can be exponentially smaller than the privacy assuming quantum input
registers. We also argue that the right notion of privacy of a communication
protocol is the one assuming classical input registers, since otherwise the
players can deviate considerably from the protocol.
