In this paper, we present two robust, thresholding based distributed algorithms for recovering sparse signal from compressed measurements. These algorithms use Lorentzian norm as the cost function which has been observed to give robustness against heavy tailed noise, e.g., impulsive noise. The first algorithm named Diffusion based Lorentzian Iterative Hard Thresholding (DLIHT) requires only gradient information whereas the second algorithm named Diffusion based Lorentzian Hard Thresholding Pursuit(DLHTP) requires solving a linear system, on top of the gradient based update. It is observed through simulations that for moderate corruptions, DLHTP converges very quickly than DLIHT to similar steady state error value. However, for higher levels of corruptions, DLIHT leads to much less steady state error than offered by DLHTP.