Social media rumours, a form of misinformation, can mislead the public and cause significant economic and social disruption. Motivated by the observation that the user network — which captures who engage with a story — and the comment network — which captures how they react to it — provide complementary signals for rumour detection, in this paper, we propose DUCK (rumour  ̲detection with  ̲user and  ̲comment networ ̲ks) for rumour detection on social media. We study how to leverage transformers and graph attention networks to jointly model the contents and structure of social media conversations, as well as the network of users who engaged in these conversations. Over four widely used benchmark rumour datasets in English and Chinese, we show that DUCK produces superior performance for detecting rumours, creating a new state-of-the-art. Source code for DUCK is available at: https://github.com/l tian678/DUCK-code.