We consider a many-to-one matching market where colleges share true preferences over students but make decisions using only independent noisy rankings. Each student has a true value $v$, but each college $c$ ranks the student according to an independently drawn estimated value $v + X_c$ for $X_c\sim \mathcal{D}.$ We ask a basic question about the resulting stable matching: How noisy is the set of matched students? Two striking effects can occur in large markets (i.e., with a continuum of students and a large number of colleges). When $\mathcal{D}$ is light-tailed, noise is fully attenuated: only the highest-value students are matched. When $\mathcal{D}$ is long-tailed, noise is fully amplified: students are matched uniformly at random. These results hold for any distribution of student preferences over colleges, and extend to when only subsets of colleges agree on true student valuations instead of the entire market. More broadly, our framework provides a tractable approach to analyze implications of imperfect preference formation in large markets.