Understanding how people form and revise their perception of risk is central to designing efficient risk communication methods, eliciting risk awareness, and avoiding unnecessary anxiety among the public. However, public responses to hazardous events such as climate change, contagious outbreaks, and terrorist threats are complex and difficult-to-anticipate phenomena. Although many psychological factors influencing risk perception have been identified in the past, it remains unclear how perceptions of risk change when propagated from one person to another and what impact the repeated social transmission of perceived risk has at the population scale. Here, we study the social dynamics of risk perception by analyzing how messages detailing the benefits and harms of a controversial antibacterial agent undergo change when passed from one person to the next in 10-subject experimental diffusion chains. Our analyses show that when messages are propagated through the diffusion chains, they tend to become shorter, gradually inaccurate, and increasingly dissimilar between chains. In contrast, the perception of risk is propagated with higher fidelity due to participants manipulating messages to fit their preconceptions, thereby influencing the judgments of subsequent participants. Computer simulations implementing this simple influence mechanism show that small judgment biases tend to become more extreme, even when the injected message contradicts preconceived risk judgments. Our results provide quantitative insights into the social amplification of risk perception, and can help policy makers better anticipate and manage the public response to emerging threats.