Clinical research networks (CRNs), made up of multiple healthcare systems each with patient data from several care sites, are beneficial for studying rare outcomes and increasing generalizability of results. While CRNs encourage sharing aggregate data across healthcare systems, individual systems within CRNs often cannot share patient-level data due to privacy regulations, prohibiting multi-site regression which requires an analyst to access all individual patient data pooled together. Meta-analysis is commonly used to model data stored at multiple institutions within a CRN but can result in biased estimation, most notably in rare-event contexts. We present a communication-efficient, privacy-preserving algorithm for modeling multi-site zero-inflated count outcomes within a CRN. Our method, a one-shot distributed algorithm for performing hurdle regression (ODAH), models zero-inflated count data stored in multiple sites without sharing patient-level data across sites, resulting in estimates closely approximating those that would be obtained in a pooled patient-level data analysis. We evaluate our method through extensive simulations and two real-world data applications using electronic health records: examining risk factors associated with pediatric avoidable hospitalization and modeling serious adverse event frequency associated with a colorectal cancer therapy. In simulations, ODAH produced bias less than 0.1% across all settings explored while meta-analysis estimates exhibited bias up to 12.7%, with meta-analysis performing worst in settings with high zero-inflation or low event rates. Across both applied analyses, ODAH estimates had less than 10% bias for 18 of 20 coefficients estimated, while meta-analysis estimates exhibited substantially higher bias. Relative to existing methods for distributed data analysis, ODAH offers a highly accurate, computationally efficient method for modeling multi-site zero-inflated count data.