Cookies are designed to enable more accurate identification and tracking of user behavior, in turn allowing for more personalized ads and better performing ad campaigns. Given the additional information that is recorded, questions related to privacy and fairness naturally arise. How does a user's consent decision influence how much the system can learn about their demographic and tastes? Is the impact of a user's consent decision on the recommender system's ability to learn about their latent attributes uniform across demographics? We investigate these questions in the context of an engagement-driven recommender system using simulation. We empirically demonstrate that when consent rates exhibit demographic-dependence, user consent has a disparate impact on the recommender agent's ability to estimate users' latent attributes. In particular, we find that when consent rates are demographic-dependent, a user disagreeing to share their cookie may counter-intuitively cause the recommender agent to know more about the user than if the user agreed to share their cookie. Furthermore, the gap in base consent rates across demographics serves as an amplifier: users from the lower consent rate demographic who agree to cookie sharing generally experience higher estimation errors than the same users from the higher consent rate demographic, and conversely for users who choose to disagree to cookie sharing, with these differences increasing in consent rate gap. We discuss the need for new notions of fairness that encourage consistency between a user's privacy decisions and the system's ability to estimate their latent attributes.