  We study the two-party communication complexity of functions with large
outputs, and show that the communication complexity can greatly vary depending
on what output model is considered. We study a variety of output models,
ranging from the open model, in which an external observer can compute the
outcome, to the XOR model, in which the outcome of the protocol should be the
bitwise XOR of the players' local outputs. This model is inspired by XOR games,
which are widely studied two-player quantum games. We focus on the question of
error-reduction in these new output models. For functions of output size k,
applying standard error reduction techniques in the XOR model would introduce
an additional cost linear in k. We show that no dependency on k is necessary.
Similarly, standard randomness removal techniques, incur a multiplicative cost
of $2^k$ in the XOR model. We show how to reduce this factor to O(k). In
addition, we prove analogous error reduction and randomness removal results in
the other models, separate all models from each other, and show that some
natural problems, including Set Intersection and Find the First Difference,
separate the models when the Hamming weights of their inputs is bounded.
Finally, we show how to use the rank lower bound technique for our weak output
models.
