Perceptive mobile networks (PMNs) were proposed to integrate sensing capability into current cellular networks where multiple sensing nodes (SNs) can collaboratively sense the same targets. Besides the active sensing in traditional radar systems, passive sensing based on the uplink communication signals from mobile user equipment may play a more important role in PMNs, especially for targets with weak electromagnetic wave reflection, e.g., pedestrians. However, without the properly designed active sensing waveform, passive sensing normally suffers from low signal to noise power ratio (SNR). As a result, most existing methods require a large number of data samples to achieve an accurate estimate of the covariance matrix for the received signals, based on which a power spectrum is constructed for localization purposes. Such a requirement will create heavy communication workload for PMNs because the data samples need to be transferred over the network for collaborative sensing. To tackle this issue, in this paper we leverage the sparse structure of the localization problem to reduce the searching space and propose an iterative sparse recovery (ISR) algorithm that estimates the covariance matrix and the power spectrum in an iterative manner. Experiment results show that, with very few samples in the low SNR regime, the ISR algorithm can achieve much better localization performance than existing methods.