  A single-sensor two-detectors system is considered where the sensor
communicates with both detectors and Detector 1 communicates with Detector 2,
all over noise-free rate-limited links. The sensor and both detectors observe
discrete memoryless source sequences whose joint probability mass function
depends on a binary hypothesis. The goal at each detector is to guess the
binary hypothesis in a way that, for increasing observation lengths, the
probability of error under one of the hypotheses decays to zero with largest
possible exponential decay, whereas the probability of error under the other
hypothesis can decay to zero arbitrarily slow. For the setting with zero-rate
communication on both links, we exactly characterize the set of possible
exponents and the gain brought up by cooperation, in function of the number of
bits that are sent over the two links. Notice that, for this setting, tradeoffs
between the exponents achieved at the two detectors arise only in few
particular cases. In all other cases, each detector achieves the same
performance as if it were the only detector in the system. For the setting with
positive communication rates from the sensor to the detectors, we characterize
the set of all possible exponents in a special case of testing against
independence. In this case the cooperation link allows Detector~2 to increase
its Type-II error exponent by an amount that is equal to the exponent attained
at Detector 1. We also provide a general inner bound on the set of achievable
error exponents. For most cases it shows a tradeoff between the two exponents.
