My take is that while private property is not directly mentioned, it is definitely implied in the Bible.
P(Presupposition) 1: the fact that you and I use pronouns in everyday speaking (my heart, your body, her kidneys, etc.) in and of itself confirms there is a sense of boundaries amongst us. Therefore, to argue against protecting private property is deeply contradictory, as you already use it via self-ownership of your body!
P2: There are differences in everyone based on relationships to property. This explains the prohibitions against murder, theft, and coveting, as these indicate relationships such as perpetrator, victim, accomplice, etc. If we were say a hivemind, then none of these commandments would be necessary as there would be no private property and thus no differences.
Conclusion: Loving and caring for your neighbor = Respecting private property
