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In this work, we study a communication scheme in which nodes transmit sufficient statistics of their observations over non-orthogonal medium access channels for distributed estimation. This scheme unifies and generalizes several multiple access schemes such as uncoded transmissions of Gaussian observations and type-based multiple access. For the exponential family of distributions, we show that the sufficient-statistic based multiple access (SSBMA) achieves the Cramer-Rao bound on the estimation error asymptotically. Further, we argue that such an optimality result only applies to the exponential family of distributions. In addition, we present a simplified and unified analysis of asymptotic distribution of estimation error for a very general class of communication schemes and estimators. The proposed analysis method reduces the computation of asymptotic estimation error to mere calculation of derivatives. With the proposed technique, we analyze the performance of various communication schemes with optimal and suboptimal estimators and with i.i.d. (independent and identically distributed) data.