Research suggests that emerging information about infection-acquired COVID-19 immunity should be interpreted with caution. The introduction of "immunity passports" that would enable people who have recovered from COVID-19 to travel freely and return to work may therefore have detrimental consequences if not managed carefully. In two studies, we examined how perceived (suspected or imagined) recovery from COVID-19, and the concept of immunity passports, influence people's intentions to engage in behaviors aimed to reduce the spread of COVID-19. We also consider ways to lessen potential negative effects. In Study 1 (N = 1604), participants asked to imagine that they had recovered from COVID-19 reported lower social distancing intentions compared to a control condition. Participants who suspected (versus imagined) that they had recovered from past infection did not report lower preventative intentions compared to the control condition, even at high levels of certainty of past infection. In Study 2 (N = 1732), introducing the idea of immunity passports also reduced social distancing intentions compared to a control condition. The latter effect was, however, attenuated when cautious information about the equivocal science on COVID-19 was also presented to participants. Participants who suspected that they had COVID-19 in the past (compared to the control condition) revealed a similar pattern of results, but only at higher levels of certainty of past infection. Caution regarding infection-acquired COVID-19 immunity and immunity passports will be crucial in the COVID-19 response. Implications for premature pandemic announcements, as well as their potential remedies, are discussed.