A growing body of work attempts to evaluate the theory of mind (ToM) abilities of humans and large language models (LLMs) using static, non-interactive question-and-answer benchmarks. However, theoretical work in the field suggests that first-personal interaction is a crucial part of ToM and that such predictive, spectatorial tasks may fail to evaluate it. We address this gap with a novel ToM task that requires an agent to persuade a target to choose one of three policy proposals by strategically revealing information. Success depends on a persuader's sensitivity to a given target's knowledge states (what the target knows about the policies) and motivational states (how much the target values different outcomes). We varied whether these states were Revealed to persuaders or Hidden, in which case persuaders had to inquire about or infer them. In Experiment 1, participants persuaded a bot programmed to make only rational inferences. LLMs excelled in the Revealed condition but performed below chance in the Hidden condition, suggesting difficulty with the multi-step planning required to elicit and use mental state information. Humans performed moderately well in both conditions, indicating an ability to engage such planning. In Experiment 2, where a human target role-played the bot, and in Experiment 3, where we measured whether human targets' real beliefs changed, LLMs outperformed human persuaders across all conditions. These results suggest that effective persuasion can occur without explicit ToM reasoning (e.g., through rhetorical strategies) and that LLMs excel at this form of persuasion. Overall, our results caution against attributing human-like ToM to LLMs while highlighting LLMs' potential to influence people's beliefs and behavior.