Compressive sensing has emerged as an important new technique in signal acquisition due to the surprising property that a sparse signal can be captured from measurements obtained at a sub-Nyquist rate. The decoding cost of compressive sensing, however, grows superlinearly with the problem size. In distributed sensor systems, the aggregate amount of compressive measurements encoded by the sensors can be substantial, and the decode cost for all the variables involved can be large. In this paper we propose a method to combine measurements from distributed sensors. With our method we can transport and store a single combined measurement set, rather than multiple sets for all sensors. We show that via source separation and joint decoding, it is possible to recover an approximate to the original signal from combined measurements using progressive reconstruction which focuses on individual sensors. This results in a reduction in the number of variables used in decoding and consequently a reduced decoding time. We show that the computed approximation to the signal can still have sufficient accuracy for target detection. We describe the combining approach and the associated progressive reconstruction, and we illustrate them with image recovery for simple target detection examples.