The theory of compressive sensing (CS) has recently been proposed as a framework for joint signal acquisition and compression by replacing the standard sample by sample measurement approach with the idea of collecting a set of random projections of the signal; it has already been successfully employed in a number of signal processing applications, e.g., medical imaging. In this paper we consider a scenario where all nodes in a distributed application must measure some signal samples and distribute them to other participants. All nodes are also required to acquire and reconstruct the signal using CS techniques. Examples of signals could be information on the computing resource availability at nodes, on the usage of communications channels between processes on different IP machines, etc. We devise an approach based on random walks to spread CS random projections to participants in a distributed application, where nodes are organized into an overlay random network. Nodes have only local visibility of neighbors provided by a common rendez-vous node, e.g., the tracker node in Bittorrent. We analyze the performance of the proposed method by means of a simple (yet accurate) analytical model describing the structure of the so called sensing matrix that is the key mathematical structure required to reconstruct the signal in CS theory. We validate our model predictions against a simulator of the system at the node and network level on different models of random networks. The model we developed can be exploited to select the parameters of the random walk and the criteria to build the sensing matrix in order to achieve successful signal recovery. We also investigate how the proposed method performs in a dynamical scenario where nodes join and leave the overlay and the signal is time varying. The analysis reveals that the method we propose is accurate, robust to node and signal dynamics, and feasible because it requires very little communication overhead and reasonable processing power.