We study distributed least-mean square (LMS) estimation problems over adaptive networks, where nodes cooperatively work to estimate and track common parameters of an unknown system. We consider a scenario where the input and output response signals of the unknown system are both contaminated by measurement noise. In this case, if standard distributed estimation is performed without considering the effect of regression noise, then the resulting parameter estimates will be biased. To resolve this problem, we propose a distributed LMS algorithm that achieves asymptotically unbiased estimates via diffusion adaptation. We analyze the performance of the proposed algorithm and provide computer experiments to illustrate its behavior.