There is no one system of property used by tribes before the modern conception of private property came to be used. And although "private property" itself is not always well-defined, it's usually defined in academic (historical/sociological) contexts as proper to modern, capitalist societies, meaning its rise & dominance was an ongoing process that's generally considered to have started circa the 15th or 16th centuries, and it only really became the dominant form of property in Europe around the 19th century. So, it's clear that the absence of private property in that sense is not the lack of rules concerning property or ownership.

Starting from that, what you call private property mostly seems to be what is often called personal property. To a degree there was almost certainly a conception of personal property, at least informally, but it doesn't necessarily follow that every person had a full suite of useful objects they considered exclusively theirs like we do today: generally useful objects like tools can be held in common or considered to be the property of an entire group. Nevertheless, obviously if one person has a personal attachment to an object, or is the only one using it regularly, and it's taken or destroyed by someone else, then that's grounds for conflict. Methods of conflict-resolution can be varied and there's no way to say from a vague fictional example how it would have happened. Probably there would have been some form of arbitration along traditional or religious lines between the people concerned. Often there were more developed rules of property for important goods that could be traded to cement alliances and so on. Cattle, land in some circumstances, and marriage (being a kind of "trade") being examples. Often in those cases even societies that we think of as quite primitive had some form of contract (verbal or written) and clear rules concerning what everyone's rights are and what happens if the agreement is broken. But everyday objects used by most people were not necessarily subject to such specific arrangements (unlike today where the law is usually pretty clear about who owns what) and problems re: their use were likely to be considered private grievances to be considered in roughly the same way as, say one person insulting the other publicly.

This is all pretty general and sadly I don't know a single anthropological source that could clear up all that. Anthropologists like Marshall Sahlins and Marcel Mauss (as well as many others) have written about the economics of such "primitive" societies and can provide examples of how "property" could function in those societies. Anthropologist and economist Karl Polanyi lays out a conception according to which pre-capitalist economies were "embedded" in non-economic aspects of society. So questions that are considered purely economic questions today were really questions of tradition, religion, family ties or clan relationships and so on. That property and rules surrounding the economy are so clearly defined and important in our societies (to the point that often family ties, religion, etc. are subject to rules re: property, inheritance, and not the other way around) is for Polanyi a case of the economy not only becoming "disembedded" from other social concerns but also expanding and dominating other spheres of social life, so that eventually everything would be "embedded" in the economy and determined by it.

It's difficult to answer such a vague questions (with a lot of assumptions about property and how human beings live) with clear and sourced statements. I hope naming a few anthropologists whose work are easy to find (and assuredly deal with related subjects) is enough.