I study how to measure information transmission when agents have misspecified priors, misinterpret signals, or update beliefs in a non-Bayesian way. I axiomatize and characterize measures of \emph{meaningful information transmission} that account for both the objective features of a signal structure and agents‘ (mis)interpretations of it. Meaningful information measures coincide with canonical ones under correct Bayesian beliefs, but otherwise extend them in a disciplined way grounded in the same underlying decision geometry. I then estimate meaningful information in three empirical applications and show that it can diverge sharply from canonical measures.