In this paper, distributed detection of sparse stochastic signals with one-bit measurements is studied. We assume that both the noise and the dominant elements in sparse signals follow the generalized Gaussian (GG) distribution. Due to constrained bandwidth/energy in sensor networks, the local sensors send binary measurements instead of analog data to the fusion center. First, we propose the locally most powerful test (LMPT) detector based on one-bit measurements for distributed detection of sparse signals. Then, we analytically derive near-optimal one-bit quantizers at the local sensor nodes using the log-concave approximation of the efficacy. Simulation results corroborate our theoretical analysis and show that, the proposed one-bit LMPT detector with analytically obtained one-bit quantizers provides detection performance which is comparable to that with numerically obtained optimal quantizers in the GG case.