What was intended by the statement 'Jews don't believe in Original Sin' is 'Jews don't believe in the ramifications of the Original Sin as described by Catholics'. Specifically, Jews don't believe that every person is born into a state of sinfulness which must then be cleansed. It is obvious of course that the physical world exists in a lowered state because of Adam's fall and that we, in turn, are affected. But individuals are born pure until they themselves sin. The biggest difference between the two opinions is that a newborn child who dies, according to Catholics goes to hell and according to Jews goes straight to Gan Eden (heaven).
I believe the misunderstanding is based on the statement 'the sins of a father will fall on their children'. According to Christians this means the state of guilt resulting from a sin. According to Jews this means two things: first that the punishment (but not guilt) will be felt by a child and second that the child will be more prone to succumbing to that sin (like the child of an alcoholic tends to be an alcoholic). Guilt is non-transferable.