...so if a friend I brought asks me for an image of a black cat I know approximately where to find it on the cat island beach

For a single image we have many bottles, the ones that do not sink end up either on cat or dog island, and we decide by counting which island got more bottles. We repeat this for many images and currents change so that bottles end up more often on the right island. Once the currents are set, if we stand in the dog/cat island, walk the beach and observe for a while we can tell from which part of the land the bottle came and what was in it