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 or to a small positive number arbitrarily slow. For the setting with positive communication rates from the sensor to the detectors and when both detectors are interested in maximizing the error exponent under the same hypothesis, 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 that shows a tradeoff between the exponents at the two detectors in most cases. When the two detectors aim at maximizing the error exponent under different hypotheses and the distribution at the Sensor is different under the two hypotheses, then we show that such a tradeoff does not exist. We propose a general scheme that allows each detector to attain the same exponent as if it was the only detector in the system. 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.