This paper gives algebraic definitions for obtaining the minimal transition and place flows of a modular Petri net from the minimal transition and place flows of its components. The notion of modularity employed is based on place sharing. It is shown that transition and place flows are not dual in a modular sense under place sharing alone, but that the duality arises when also considering transition sharing. As an application, the modular definitions are used to give compositional definitions of transition and place flows of models in a subset of the Calculus of Biochemical Systems.