This paper focuses on passive emitter localization using unmanned aerial vehicle (UAV) clusters. UAV clusters are regarded as an excellent solution for the rapid and flexible construction of wireless communication networks in emergency scenarios. The direct position determination (DPD) algorithms have been proved to outperform the two-step algorithms, but they introduce significant computational and communication overhead, which is intolerable for resource-constrained UAV clusters. To address the challenging problem of high-quality passive emitter localization with limited communication and computational overhead, we propose a deep neural network (DNN)-aided distributed DPD method for UAV cluster. Different from most existing DPD algorithms, a novel distributed computing strategy, cost function and corresponding iterative algorithm are proposed to avoid expensive global search in the region of interest. Simulation results demonstrate that the proposed method considerably reduces communication and computational overhead, while maintaining similar positioning performance to the traditional DPD in the low signal-to-noise ratio (SNR) regime.