In many distributed sensing applications with limited resources, it is likely that only a few agents will have valuable information at any given time. Therefore it is important to ensure that the resources are spent on communicating valuable information from informative agents. This paper presents communication-efficient distributed sensing algorithms that avoid network cluttering by having only agents with high Value of Information (Vol) broadcast their measurements to the network, while others censor themselves. A novel contribution of the presented distributed estimation algorithm is the use of an adaptively adjusted Vol threshold to determine which agents are informative. This adaptation enables the team to better balance between the communication cost incurred and the long-term accuracy of the estimation. Theoretical results are presented establishing the almost sure convergence of the communication cost and estimation error for distributions in the exponential family. Furthermore, validation through numerical simulations and real datasets shows that the new Vol-based algorithms can yield improved parameter estimates than those achieved by previously published hyperparameter consensus algorithms while incurring only a fraction of the communication cost. Published by Elsevier Ltd.