We study distributed estimation of a source corrupted by an additive Gaussian noise and observed by sensors which are connected to a fusion center with unknown orthogonal (parallel) flat Rayleigh fading channels. The fading communication channels are estimated with training. Subsequently, source estimation given the channel estimates and transmitted sensor observations is performed. We consider a setting where the estimated channels are fed-back to the sensors for optimal power allocation which leads to a threshold behavior of sensors with bad channels being unused (inactive). We also show that at least half of the total power should be used for training. Simulation results corroborate our analytical findings.