In this paper, we propose a new class of bivariate point process models to model the activity patterns of social media users. The proposed class of models has the flexibility to accommodate the complex behaviors of modern social media users and to provide straightforward insight into users' online content generating behavior. A composite likelihood approach and a composite likelihood EM procedure are developed to overcome the challenges in parameter estimation. We show the consistency and asymptotic normality of the maximum composite likelihood estimator. We apply our proposed method to President Donald Trump's Twitter data and uncover changes in various aspects of his tweeting behavior during the presidential campaign and the presidency. Moreover, we apply our method to a large scale social media data and find interesting subgroups of users with distinct behaviors. Additionally, we discuss the effect of social ties on a user's online content generating behavior.