It is challenging for survey researchers to investigate sensitive topics due to concerns about socially desirable responding (SDR). The susceptibility to social desirability bias may vary not only between individuals (e.g., different perceptions about social norms) but also within individuals (e.g., perceived sensitivity of different items). Thus, controlling for SDR is particularly challenging when analyzing multidimensional constructs that are measured via multiple groups of items with varying degrees of sensitivity. In this research, we address this challenge using a combination of a randomized response (RR) approach for data collection and a multiscale item response theory (IRT) model for data analysis. While the RR approach protects the anonymity of respondents at the item level, the multiscale IRT approach accounts for the multidimensional nature of the construct and explicitly models the item-level differences in the measurement of its dimensions. We empirically demonstrate the benefits of the model using a multidimensional self-report instrument for the assessment of academic misconduct of university students. Based on an experiment with random assignment, our results uncover considerable differences in the perceived sensitivity, both between the construct dimensions and between their measurement items. These findings support the view that individuals engage in SDR to varying degrees depending on the perceived sensitivity of the specific items and groups of items. In contrast, a social desirability scale that treats SDR as a stable personality trait is not found to capture meaningful differences in response style. Finally, we show how structural models can be incorporated into the framework to link the latent construct’s dimensions to individual-level explanatory variables.