It seems plausible that successfully communicating with our peers requires entertaining the same thoughts as they do. We argue that this view is incompatible with other, independently plausible principles of thought individuation. Our argument is based on a puzzle inspired by the Kripkean story of Peter and Paderewski: having developed several variations of the original story, we conclude that understanding and communication cannot be modeled as a process of thought transfer between speaker and hearer. While we are not the first to reach this conclusion, the significance of our argument lies in the fact that it only relies on widely accepted premises, without depending on any especially controversial theory of mental and linguistic content. We conclude by drawing out the implications of that conclusion: if communication and understanding do not require thought identity, then one important motivation for the postulation of inter-personally shared thoughts is undercut.