Nowadays, news apps have taken over the popularity of paper-based media, providing a great opportunity for personalization. Recurrent Neural Network (RNN)-based sequential recommendation is a popular approach that utilizes users' recent browsing history to predict future items. This approach is limited that it does not consider the societal influences of news consumption, i.e., users may follow popular topics that are constantly changing, while certain hot topics might be spreading only among specific groups of people. Such societal impact is difficult to predict given only users' own reading histories. On the other hand, the traditional User-based Collaborative Filtering (UserCF) makes recommendations based on the interests of the "neighbors", which provides the possibility to supplement the weaknesses of RNN-based methods. However, conventional UserCF only uses a single similarity metric to model the relationships between users, which is too coarse-grained and thus limits the performance. In this paper, we propose a framework of deep neural networks to integrate the RNN-based sequential recommendations and the key ideas from UserCF, to develop Collaborative Sequential Recommendation Networks (CSRNs). Firstly, we build a directed co-reading network of users, to capture the fine-grained topic-specific similarities between users in a vector space. Then, the CSRN model encodes users with RNNs, and learns to attend to neighbors and summarize what news they are reading at the moment. Finally, news articles are recommended according to both the user's own state and the summarized state of the neighbors. Experiments on two public datasets show that the proposed model outperforms the state-of-the-art approaches significantly.