Adam Yu has written a wonderfully titled essay, "The Coase Theorem Meets Moneyball." It won the Milton Friedman prize at the Chicago Graduate School of Business, and it is available on-line here [free registration required to see the entire essay]. The essay has a very clear statement, applying the strong version of Coase to the change from the reserve clause to free agency:
Prior to the inception of free agency in 1976, the property rights of players were solely assigned to the team. This policy, known as the reserve clause, meant that barring a trade or sale of player rights, individuals could only play for their drafted team. With the advent of free agency, players who had acquired a requisite amount of service time were free to the highest bidder. Free agency thereby signified the transfer of property rights from teams (owners) to players. Owners vehemently resisted free agency, fearing the disproportionate flow of premium talent towards the larger baseball markets. According to Coase however, free agency's transfer of property rights would leave the allocation of players unaffected. Even under the reserve clause then, it was argued that the better players from smaller market clubs were peddled to the larger market clubs anyway.
This opening provides an excellent introduction. But then the essay argues that high transaction costs, in the form of high salaries, keep small market teams from being competitive with the large market teams:
However, the best underpaid talent will not be underpaid for long, nor will they be with their original team. In the end, Coase was right again in theory; but free agency created a transaction cost in the form of rising salaries that altered the distribution of major league baseball talent.
Oops. Rising salaries are a result of the realignment of property rights; they aren't transaction costs.
To the extent that results are different under free agency than they would have been under the reserve clause, one possible explanation is that transaction costs are higher now than they were then [which seems plausible given stories we read about the negotiations between owners and agents]; but that would mean we should see more competitive balance now than we did before free agency, which the author seems to deny.
An alternative explanation that seems plausible is that many of the current property rights owners (the players) have utility functions with different arguments and weights than the previous property rights owners (the team owners). We still get an efficient allocation of scarce resources, but it is different, and this is the weak version of the Coase Theorem.