There has been extensive research on community detection in directed and bipartite networks. However, these studies often fail to consider the popularity of nodes in different communities, which is a common phenomenon in real-world networks. To address this issue, we propose a new probabilistic framework called the Two-Way Node Popularity Model (TNPM). The TNPM also accommodates edges from different distributions within a general sub-Gaussian family. We introduce the Delete-One-Method (DOM) for model fitting and community structure identification, and provide a comprehensive theoretical analysis with novel technical skills dealing with sub-Gaussian generalization. Additionally, we propose the Two-Stage Divided Cosine Algorithm (TSDC) to handle large-scale networks more efficiently. Our proposed methods offer multi-folded advantages in terms of estimation accuracy and computational efficiency, as demonstrated through extensive numerical studies. We apply our methods to two real-world applications, uncovering interesting findings.