What effect does private property seem to have on the position of women in society?
Though specifics of access to power and access to private property differ wildly through human history, one can say safely that a woman's access to private property directly correlates with her access to power. This is readily evident within the history of the women's empowerment movement over the course of the last two hundred years in the United States. As women have gained increased access to private property, so too have they seen increased access to power.
Until the early nineteenth century, women's lives were dominated by laws of coverture. As a child and young adult, a woman was a dependent of her father's. When she married, a woman was a dependent of her husband's. Should she be widowed, the law stated she and her possessions were dependents of her son's. So, unless a woman had a trust and was able to live independently off its funds, women had negligible opportunities to legally own property. Early in the nation's history, there was a clear relationship between lack of property ownership and lack of political participation.
When women earned the right to vote in 1919, they saw their power and access to property change. Though women's empowerment has been slow and marked with episodes of regression, women have increased their share of property and wealth possession in the US dramatically in the intermittent hundred years. Interestingly, their political participation has not increased at the same rate. While there is definitely positive change in both property ownership and political participation, it is unclear exactly to what extent one affects the other.
At the very least, owning private property gives a woman the "power of the purse," meaning she has the economic power, in a capitalist society, to decide how she spends her money.
There is no single answer to this question as the relationship of women to private property has varied across different periods and cultures. The first way to think about the issue is to look at the rules applying to gender and property ownership across different cultures.
In some cultures, such as Britain before the passage of the Married Women's Property Act of 1882, all property of women was transferred to their husbands on marriage. This very much reduced the freedom and independence of women.
In ancient Rome, women retained their dowries after marriage and could initiate divorces, returning to their fathers with dowry intact, something that made Roman matrons more powerful than their Greek counterparts.
Next, the notion of primogeniture, inheritance of property by the eldest male, also would impoverish women.
Marxist feminist critics argue that private property is the root of all forms of inequality and that to abolish private property would reduce gender inequality.