Mental illness is a global health problem, but access to mental healthcare resources remain poor worldwide. Online peer-to-peer support platforms attempt to alleviate this fundamental gap by enabling those who struggle with mental illness to provide and receive social support from their peers. However, successful social support requires users to engage with each other and failures may have serious consequences for users in need. Our understanding of engagement patterns on mental health platforms is limited but critical to inform the role, limitations, and design of these platforms. Here, we present a large-scale analysis of engagement patterns of 35 million posts on two popular online mental health platforms, TalkLife and Reddit. Leveraging communication models in human-computer interaction and communication theory, we operationalize a set of four engagement indicators based on attention and interaction. We then propose a generative model to jointly model these indicators of engagement, the output of which is synthesized into a novel set of eleven distinct, interpretable patterns. We demonstrate that this framework of engagement patterns enables informative evaluations and analysis of online support platforms. Specifically, we find that mutual back-and-forth interactions are associated with significantly higher user retention rates on TalkLife. Such back-and-forth interactions, in turn, are associated with early response times and the sentiment of posts.