We introduce in this paper a new fully distributed particle filter algorithm, referred to as the Random Exchange Diffusion Particle Filter (ReDif-PF), which is based on random information dissemination and, unlike previous consensus-based approaches, does not require iterative inter-node communication between measurement arrivals. The proposed algorithm is applied to track a moving emitter using a network of received-signal-strength (RSS) sensors. In a simulated scenario with a partially connected network, the proposed ReDif-PF outperformed an alternative suboptimal tracker that assimilates local neighboring measurements only. Compared to a broadcast-based filter that mimics the optimal centralized tracker and to its equivalent fully distributed consensus-based implementations, ReDif-PF showed a degradation in steady-state RMS error performance. However, ReDif-PF is better suited for real-time applications since it requires much lower bandwidth than the consensus-based filters and also has a lower computational cost.