There are many potential points of failure, at least as regards the cows.
Reelya has neatly summed up the "Owns cow" angle, but if we throw in things like "Milking/butchering the cow", and then "Distribution of meat/milk", we get more opportunities to extract personal gain from the communal resource.
EG-- "OK, the cows are community property. Do not take all the milk for yourself/do not butcher the cow yourself."
What stops somebody from doing exactly those things? Sure, the cow belongs to everyone, but when you milk it, all the milk ends up in one bucket. Who's bucket is it, and who ensures the distribution?
Likewise, when you butcher the cow, all the meat ends up in somebody's butcher shop. Who's shop, and who oversees distribution?
"Assigned persons!" somebody may exclaim--
"The assigned milker milks the cow, and puts the milk into the public dispensary."
and
"The assigned butcher kills the cow and cuts it up, then presents the meat to the public."
What stops the milker from holding back some of the milk themselves for personal enrichment, away from public hands?
What stops the butcher from holding back some of the more choice cuts of meat in a similar fashion?
Then--- there is the distribution angle--
"Well, we assign monitors to watch the milker and the butcher, so they dont cheat the public!"
OK, now we have soviet russia. Who watches the watchers? Who ensures that the distribution is fair, and that it is not being imbalanced in favor of the ones doing the distribution?
This is always a losing game. There really is no solution to tragedy of the commons, aside from universal condemnation of greed in the community. The consequences (and fear of consequences) imposed by the rest of the community are really the only thing that can keep the tragedy of the commons at bay. Once the community is shorn of the power to impose direct penalties to the parties engaged in exploitation, there is no stopping the plunder of the commons.