Summary: In estimating the proportion \(\theta_A\) of people in a given community bearing a sensitive characteristic \(A\), in order to protect the respondent’s privacy, various techniques of generating randomized response (RR) rather than direct response are available in the literature. But the theory concerning them is developed only for samples selected by ‘simple random sampling’ (SRS) ‘with replacement’ or at most by SRS without replacement method. Illustrating two such RR devices we show how an estimator along with an estimated measure of its error may be developed if the sample of persons may be drawn adopting a complex survey design involving unequal selection probabilities with or without replacement.