Abstract: Linear $L_1$-regularized models have remained one of the simplest and most effective tools in data science. Over the past decade, screening rules have risen in popularity as a way to reduce the runtime for producing the sparse regression weights of $L_1$ models. However, despite the increasing need of privacy-preserving models for data analysis, to the best of our knowledge, no differentially private screening rule exists. In this paper, we develop the first differentially private screening rule for linear and logistic regression. In doing so, we discover difficulties in the task of making a useful private screening rule due to the amount of noise added to ensure privacy. We provide theoretical arguments and experimental evidence that this difficulty arises from the screening step itself and not the private optimizer. Based on our results, we highlight that developing an effective private $L_1$ screening method is an open problem in the differential privacy literature.