A wireless sensor network performs spatial inference on a physical phenomenon of interest. The areas in which this phenomenon exhibits interesting or anomalous behavior are identified whilst controlling false positives. We expand our previous work based on multiple hypothesis testing (MHT) and local false discovery rates to save energy and reduce spectrum use. The number of transmissions from sensors producing uninformative statistics are reduced by introducing censoring for MHT that imposes a communication rate constraint while maintaining the desired performance. Two novel methods are proposed. As our numerical experiments demonstrate, both approaches reduce the number of transmissions while maintaining false discovery rate control. In addition, one method allows to either define a fixed number of total transmissions or to trade thenumber of transmissions off against the achieved detection power.