Hearing aids are electronic, battery-operated sensing devices which aim at compensating various kinds of hearing impairments. Recent advances in low-power electronics coupled with progresses made in digital signal processing offer the potential for substantial improvements over state-of-the-art systems. Nevertheless, efficient noise reduction in complex listening scenarios remains a challenging task, partly due to the limited number of microphones that can be integrated on such devices. We investigate the noise reduction capability of hearing instruments that may exchange data by means of a rate-constrained wireless link and thus benefit from the signals recorded at both ears of the user. We provide the necessary theoretical results to analyze this collaboration mechanism under two different coding strategies. The first approach takes full benefit of the binaural correlation, while the second neglects it, since binaural statistics are difficult to estimate in a practical setting. The gain achieved by collaborating hearing aids as a function of the communication bit rate is then characterized, both in a monaural and a binaural configuration. The corresponding optimal rate allocation strategies are computed in closed form. While the analytical derivation is limited to a simple acoustic scenario, the latter is shown to capture many of the features of the general problem. In particular, it is observed that the loss incurred by coding schemes which do not consider the binaural correlation is rather negligible in a very noisy environment. Finally, numerical results obtained using real measurements corroborate the potential of our approach in a realistic scenario.