This paper proposes a distributed multiple constraints generalized side lobe canceler algorithm for speech enhancement in a wireless acoustic sensor network comprising fully connected microphone nodes. The algorithm is designed for reverberant environments with both desired and competing speakers. It obtains significant communication bandwidth reduction by performing local beamforming at each node rather than broadcasting all microphone signals. Each node processes its own microphone signals along with transmitted signals from other nodes. The generalized side lobe canceler form allows adaptation of the beamformer during non-speech segments and relaxes requirements of other distributed algorithms to re-estimate source positions after each iteration. Experimental results validate the theoretical analysis for both narrowband and wideband speech signals.