AI and NLP publication venues have increasingly encouraged researchers to reflect on possible ethical considerations, adverse impacts, and other responsible AI issues their work might engender. However, for specific NLP tasks our understanding of how prevalent such issues are, or when and why these issues are likely to arise, remains limited. Focusing on text summarization—a common NLP task largely overlooked by the responsible AI community—we examine research and reporting practices in the current literature. We conduct a multi-round qualitative analysis of 333 summarization papers from the ACL Anthology published between 2020–2022. We focus on how, which, and when responsible AI issues are covered, which relevant stakeholders are considered, and mismatches between stated and realized research goals. We also discuss current evaluation practices and consider how authors discuss the limitations of both prior work and their own work. Overall, we find that relatively few papers engage with possible stakeholders or contexts of use, which limits their consideration of potential downstream adverse impacts or other responsible AI issues. Based on our findings, we make recommendations on concrete practices and research directions.