I think I understand the main implication of the Coase Theorem—that the outcome will be Pareto efficient regardless of the initial allocation of property rights, provided transaction costs are zero, but I am not sure whether the outcome is the same from the perspective of each party.
As far as I can see the initial allocation of property rights has a major impact on the costs to each party because it determines which party ends up bearing the costs—either of eliminating the cause of the externality or by compensating the victim for it.
I made a decision-tree to illustrate the point based on the historic case of Bryant v. Lefever that Coase described in The Problem of Social Cost (page 11):
Can someone confirm this? In this sense, the deliberations of the judges in deciding rights are not inconsequential, even if they do not affect the optimum solution from a societal perspective.