We introduce a differentially private method to measure nonlinear correlations (as point estimates) between sensitive data hosted across two entities. We then build up on this method to introduce π-test, a privacy-preserving hypothesis test of statistical independence between data distributed across two parties. We consider a one way communication paradigm where an intermediate computation is privatized and released by only one party. The other party that receives this communication, performs further computations over it to obtain the final estimate (or output of test). We provide utility bounds of our private point estimator of nonlinear correlation. Our method is based on the connections between the Johnson-Lindenstrauss transform and differential privacy to release statistics of distances, graph Laplacians and directional variances. We therefore exploit random projections in different forms to provide privacy mechanisms and in their analysis. The important measure of nonlinear correlation that we privatize is that of distance correlation. π-test relies on a private estimate for the mean of pairwise distances in addition to the private distance correlation estimate. We establish both additive and multiplicative error bounds on the utility of our differentially private test-statistic used for independence testing. We believe this work will find applications in a variety of distributed hypothesis testing settings involving sensitive data