Information flow security properties were defined some years ago (see, e.g., the surveys \cite{FG01,Ry01}) in terms of suitable equivalence checking problems. These definitions were provided by using sequential models of computations (e.g., labeled transition systems \cite{GV15}), and interleaving behavioral equivalences (e.g., bisimulation equivalence \cite{Mil89}). More recently, the distributed model of Petri nets has been used to study non-interference in \cite{BG03,BG09,BC15}, but also in these papers an interleaving semantics was used. We argue that in order to capture all the relevant information flows, truly-concurrent behavioral equivalences must be used. In particular, we propose for Petri nets the distributed non-interference property, called DNI, based on {\em branching place bisimilarity} \cite{Gor23b}, which is a sensible, decidable equivalence for finite Petri nets with silent moves. Then we focus our attention on the subclass of Petri nets called {\em finite-state machines}, which can be represented (up to isomorphism) by the simple process algebra CFM \cite{Gor17}. DNI is very easily checkable on CFM processes, as it is compositional, so that it does does not suffer from the state-space explosion problem. Moreover, we show that DNI can be characterized syntactically on CFM by means of a type system.