We investigate the problem of distributed sensors' failure detection in networks with a small number of defective sensors, whose measurements differ significantly from neighboring sensor measurements. Defective sensors are represented by non-zero values in binary sparse signals. We build on the sparse nature of the binary sensor failure signals and propose a new distributed detection algorithm based on Group Testing (GT). The distributed GT algorithm estimates the set of defective sensors from a small number of linearly independent binary messages exchanged by the sensors. The distributed GT algorithm uses a low complexity distance decoder that is robust to noisy messages. We first consider networks with only one defective sensor and determine the minimal number of linearly independent messages needed for detection of the defective sensor with high probability. We then extend our study to the detection of multiple defective sensors by modifying appropriately the message exchange protocol and the decoding procedure. We show through experimentation that, for small and medium sized networks, the number of messages required for successful detection is actually smaller than the minimal number computed in the analysis. Simulations demonstrate that the proposed method outperforms methods based on random walk measurements collection in terms of detection performance and convergence rate. Finally, the proposed method is resilient to network dynamics due to the effective gossip-based message dissemination protocol.