We present the theoretical framework and the results of a pilot survey conducted in Calabria, a region in the south of Italy, to investigate the prevalence of two sensitive characteristics, namely induced abortion among foreign women residing in this region, and irregular immigrant status. Collecting data on these two attributes by means of traditional survey modes typically produces underestimates of the diffusion of the phenomena due to the stigmatizing nature of the investigated topics. In order to overcome this problem, we employ an alternative data collection method known as the Randomized Response Technique. In particular, we focus on the implementation of the crossed model recently proposed by Lee et al. (Stat Probab Lett 83:399–409, 2013) to estimate two sensitive characteristics and some related measures of association.