A detection system with a single sensor and $\mathsf{K}$ detectors is considered, where each of the terminals observes a memoryless source sequence and the sensor sends a common message to all the detectors. The communication of this message is assumed error-free but rate-limited. The joint probability mass function (pmf) of the source sequences observed at the terminals depends on an $\mathsf{M}$-ary hypothesis $(\mathsf{M} \geq \mathsf{K})$, and the goal of the communication is that each detector can guess the underlying hypothesis. Each detector $k$ aims to maximize the error exponent under hypothesis $k$, while ensuring a small probability of error under all other hypotheses. This paper presents an achievable exponents region for the case of positive communication rate, and characterizes the optimal exponents region for the case of zero communication rate. All results extend also to a composite hypothesis testing scenario.