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In a cooperative broadcast scenario, a group of nodes in a network aims to reconstruct a common message. In this paper, we present a new algorithm for distributed consensus-based estimation in such scenarios. Possible applications comprise mobile communication systems and sensor networks. Starting with a least squares estimation problem, the algorithm is developed using techniques from optimization theory. The required communication effort for parallel implementation in a resource-constrained network is estimated and compared to existing approaches. We show that the proposed algorithm requires fewer iterations and a reduced communication overhead per iteration while keeping the estimation accuracy. A modification of the algorithm based on an approximation is presented, which reduces the communication effort even further. All results are corroborated by computer simulations considering different system parameters.