News recommendation (NR) is essential for online news services. Existing NR methods typically adopt a news-user representation learning framework, facing two potential limitations. First, in news encoder, single candidate news encoding suffers from an insufficient semantic information problem. Second, existing graph-based NR methods are promising but lack effective news-user feature interaction, rendering the graph-based recommendation suboptimal. To overcome these limitations, we propose dual-interactive graph attention networks (DIGAT) consisting of news- and user-graph channels. In the news-graph channel, we enrich the semantics of single candidate news by incorporating the semantically relevant news information with a semantic-augmented graph (SAG). In the user-graph channel, multi-level user interests are represented with a news-topic graph. Most notably, we design a dual-graph interaction process to perform effective feature interaction between the news and user graphs, which facilitates accurate news-user representation matching. Experiment results on the benchmark dataset MIND show that DIGAT outperforms existing news recommendation methods. Further ablation studies and analyses validate the effectiveness of (1) semantic-augmented news graph modeling and (2) dual-graph interaction.