Epidemiologic studies frequently use risk ratios to quantify associations between exposures and binary outcomes. When the data are physically stored at multiple data partners, it can be challenging to perform individual-level analysis if data cannot be pooled centrally due to privacy constraints. Existing methods either require multiple file transfers between each data partner and an analysis center (e.g., distributed regression) or only provide approximate estimation of the risk ratio (e.g., meta-analysis). Here we develop a practical method that requires a single transfer of eight summary-level quantities from each data partner. Our approach leverages an existing risk-set method and software originally developed for Cox regression. Sharing only summary-level information, the proposed method provides risk ratio estimates and confidence intervals identical to those that would be provided - if individual-level data were pooled - by the modified Poisson regression. We justify the method theoretically, confirm its performance using simulated data, and implement it in a distributed analysis of COVID-19 data from the U.S. Food and Drug Administration's Sentinel System.