Network alignment aims at finding the correspondence of nodes across different networks, which is significant for many applications, e.g., fraud detection and crime network tracing across platforms. In practice, however, accessing the topological information of different networks is often restricted and even forbidden, considering privacy and security issues. Instead, what we observed might be the event sequences of the networks' nodes in the continuous-time domain. In this study, we develop a coupled neural point process-based (CPP) sequence modeling strategy, which provides a solution to privacy-preserving network alignment based on the event sequences. Our CPP consists of a coupled node embedding layer and a neural point process module. The coupled node embedding layer embeds one network's nodes and explicitly models the alignment matrix between the two networks. Accordingly, it parameterizes the node embeddings of the other network by the push-forward operation. Given the node embeddings, the neural point process module jointly captures the dynamics of the two networks' event sequences. We learn the CPP model in a maximum likelihood estimation framework with an inverse optimal transport (IOT) regularizer. Experiments show that our CPP is compatible with various point process backbones and is robust to the model misspecification issue, which achieves encouraging performance on network alignment. The code is available at https://github.com/Dixin-s-Lab/CNPP.