The rates of mental illness, especially anxiety and depression, have increased greatly since the start of the COVID-19 pandemic. Traditional mental illness screening instruments are too cumbersome and biased to screen an entire population. In contrast, smartphone call and text logs passively capture communication patterns and thus represent a promising screening alternative. To facilitate the advancement of such research, we collect and curate the DepreST Call and Text log (DepreST-CAT) dataset from over 365 crowdsourced participants during the COVID-19 pandemic. The logs are labeled with traditional anxiety and depression screening scores essential for training machine learning models. We construct time series ranging from 2 to 16 weeks in length from the retrospective smartphone logs. To demonstrate the screening capabilities of these time series, we then train a variety of unimodal and multimodal machine and deep learning models. These models provide insights into the relative screening value of the different types of logs, lengths of log time series, and classification methods. The DepreST-CAT dataset is a valuable resource for the research community to model communication patterns during the COVID-19 pandemic and further the development of machine learning algorithms for passive mental illness screening.