This varies with the culture - in ancient Egypt and Babylon the temples claimed ownership of the land, but the peasants had rights to their plots. You will find something similar in all very hierarchical systems. If the society is flatter, ownership may be communal, with the individual plots assigned annually by the village elders, or by some heredity rule.
Note that in none of the above societies could the peasant alienate (sell) the land. Similar societies continued to exist down to and well into the first millennium BC.
OTOH, within the urban environment the rental, sale, and purchase of land and buildings, by individuals, is well documented before 1500 BC.
With the introduction of coinage, in the 6th century 600 BC, small scale transactions were facilitated, and land holdings could be monetized.
As a result you see the development of a "middle class" of small land holders, often with mortgages, marked by stones on the boundaries of the land to indicated the debt. Some even borrowed against their own person; indeed, money is the source of many evils!
So if they could keep out of debt, and pay their taxes to the state, these middle class landowners could thrive, as they did at Athens and Latium. The tendency, of course, is for the person with more capital to buy up more land, concentrating it in the hands of the more wealthy. But even in the late days of the Roman Empire, there were still plenty of small free-holders in Italy.
It requires peace and prosperity for the free-holder to exist; in times of trouble they would sell out, or bond themselves to the local lord, who offered protection via his private militia. This was a common event in late Roman times, and even after.
This is not really an answer for your question, but perhaps a guide as to how to search the histories for an answer.