In the case of externalities, does the initial allocation of rights affect the final outcome from the perspective of each party? 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.
Yes you're correct.
Stigler's &quot;Coase Theorem&quot; merely asserts that if transaction costs are zero, then the initial allocation of rights will not affect the total size of the economic pie, but may affect the distribution of the pie.
Two examples:
Example 1. Profit &gt; Damage.
A producer $X$ producing widgets earns \$3 in profits but causes \$1 of pollution damage to a neighbor $Y$.

If $X$ has the right to pollute, then the widgets will be produced anyway, and $Y$ will simply suffer \$1 of damage (while $X$ earns \$3 in profits).
If $X$ has no right to pollute, then $X$ has to compensate $Y$ \$1, but the widgets will be produced anyway, and $Y$ suffers no damage, while $X$'s profits are reduced to \$2.

In either case, the widgets are produced, the pollution damage is done, and the size of the economic pie is the same. In particular, the net addition to the economic pie is \$3 - \$1 = \$2.
However, the distribution of the pie differs: In Case 1, $X$ enjoys +\$3 while $Y$ suffers -\$1. In Case 2, $X$ enjoys only +\$2 while $Y$ is not affected (+\$0).
Example 2. Profit &lt; Damage.
The profits remain \$3, but now the pollution damage is $\color{red}{\\\$4}$.

If the producer $X$ has the right to pollute, then the neighbor $Y$ will pay $X$ between \$3 and \$4 to not produce the widgets.
If $X$ has no right to pollute and must compensate for any damage, then $X$ will not produce the widgets.

In either case, the widgets are not produced, the pollution damage is not done, and the size of the economic pie is the same.
However, the distribution of the pie differs: In Case 1, $Y$ transfers to $X$ some amount between +\$3 and +\$4. In Case 2, no widgets are produced, no transfers are made, $Y$ suffers no damage, and $X$ fails to enjoy any profit.