We propose some theoretical and empirical advances by supplying the methodology for analyzing the factors that influence two sensitive variables when data are collected by randomized response (RR) survey modes. First, we provide the framework for obtaining the maximum likelihood estimates of logistic regression coefficients under the RR simple and crossed models, then we carry out a simulation study to assess the performance of the estimation procedure. Finally, logistic regression analysis is illustrated by considering real data about cannabis use and legalization and about abortion and illegal immigration. The empirical results bring out certain considerations about the effect of the RR and direct questioning survey modes on the estimates. The inference about the sign and the significance of the regression coefficients can contribute to the debate on whether the RR approach is an effective survey method to reduce misreporting and improve the validity of analyses.