When differentially private statistics are used for decision-making for numerous units (e.g., jurisdictions), some units can be effectively randomly assigned to interventions, even in the presence of an ostensibly deterministic policy. Here we develop methods for using this randomness for causal inference about the effects of those interventions. In particular, we characterize deterministic policies applied to differentially private data as corresponding to stochastic policies applied to the original private data. We develop inverse-probability-weighted estimators, which make use of the probability with which a unit is assigned to treatments by the privacy mechanism, and versions of these estimators that are themselves differentially private. These estimators can have substantial advantages over other methods, such as regression discontinuity analyses, that avoid using these probabilities of assignment. Using the U.S. Census Bureau's American Community Survey, we illustrate a potential application to how translated voter information affects voter turnout.