In a world where everyone clings to Christian love and understanding, and levels of trust are high, property rights aren't really necessary. The tragedy of the commons doesn't exist, everyone does the "right" thing.
However, in a fallen world where few people know God and His love, property rights are a necessary starting point for organizing economic activity. They are clearly approved by God (Leviticus 25, etc.)
In the Piper book Momentary Marriage, I find this quote on divorce helpful:
“there are laws in the Old Testament that are not expressions of God’s will for all time, but expressions of how best to manage sin in a particular people at a particular time. Divorce is never commanded and never instituted in the Old Testament. But it was permitted and regulated—like polygamy was permitted and regulated, and like certain kinds of slavery were permitted and regulated And Jesus says here that this permission was not a reflection of God’s ideal for his people; it was a reflection of the hardness of the human heart. “Because of your hardness of heart he wrote you this commandment.”
You can see God's commands regarding the enforcement of property rights as similar to this. That makes Jesus' words in the New Testament more powerful-- give to all who ask, and do not turn from whoever wishes to borrow from you.
One microeconomics textbook example of the importance of property rights is that of a particular university that adopted a bike-sharing program. At the end of the school year, the community bikes were all either stolen or in disrepair. "That which is owned by everyone is owned by no one." However, there are counter examples of similar programs that work fairly well. In some cases they work because incentives are aligned with a property/price mechanism--it works just like the textbook says it should. But in other cases it just seems to be that the participants behave better, they show a love and respect for each other even in the absence of property rights. People in the community maintain the bikes, even though they don't have to.
This doesn't eliminate the problem of scarcity as some claim (to be addressed in a future post) but it does show that in a God-fearing, loving community property rights don't have to be absolute. If everyone is giving up their rights then we can trust each other to do the "right" things.