A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider Poisson systems over non-discrete, non-compact, locally compact Polish groups, and we prove by construction all Poisson systems over such a group are finitarily isomorphic, producing examples of isomorphisms for non-amenable group actions. As a corollary, we prove Poisson systems and products of Poisson systems are finitarily isomorphic. For a Poisson system over a group belonging to a slightly more restrictive class than above, we further prove it splits into two Poisson systems whose intensities sum to the intensity of the original, generalizing the same result for Poisson systems over Euclidean space proved by Holroyd, Lyons, and Soo.