In many distributed sensing applications it is likely that only a few agents will have valuable information at any given time. Since wireless communication between agents is resource-intensive, it is important to ensure that the communication effort is focused 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 (VoI) 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 VoI 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 to zero for distributions in the exponential family. Furthermore, validation through real datasets shows that the new VoI-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.