First of all, it depends on what you mean by emancipation. There are two possible measures of emancipation, neither of which in my opinion is particularly good. One of them is achieved through political rights, and the other through wage earning.
Women did not gain access to suffrage in Western societies simply because migrant women’s labour freed them to become politically active. After all, women’s suffrage was not limited, over the course of the 20th century, to women employing domestic servants. So talking about political emancipation in most countries today is not so obviously compromised by the the fact that migrant women are working certain jobs. Migrant women are denied access to suffrage not because they are women but because they, along with migrant men, often have no access to permanent residency or citizenship.
The vast majority of working women are not highly paid, highly educated mothers
There’s been a great deal of discussion recently about how highly educated, employed women in Europe and north America depend on migrant caregivers so they can take jobs outside their homes. I think this is a real concern but its impact on women’s emancipation generally is probably an exaggerated one. The vast majority of working women are not highly paid, highly educated mothers nor do the majority of working women have domestic servants of any kind. Domestic servants are not as commonly employed in wealthy nations of the world today as they were, for example, in 19th century Britain, when even modest, lower middle class families employed domestic servants. I don’t think we should reach conclusions about enhanced or higher levels of emancipation for women in western societies based on the fact that a small population of very well educated and wealthy women employ migrant women as servants in their homes.
In a more structural sense of course most men and women in western societies depend on the care work of women earning low wages in many places, including commercial kitchens, hospitals, childcare, and care facilities for the elderly. That might be a foundation for the emancipation of middle class and native women, but I’m not completely sure they benefit more than men in such societies.
So if the question is, whether the emancipation of women in western societies is built on, or requires, the exploitation of other migrant women, I hope I’ve given at least the beginnings of an answer.
There’s another question, though, about what or who, if anyone, has stepped in to replace the unpaid labour women have long done in households, as larger numbers of women earn wages outside the home. In some countries, lower class women – who may or may not be migrants –have taken up that burden. In some cases fathers, husbands and children do more housework. But in other countries women’s work has been socialised and commercialised; women earn wages and are covered by labour laws, for example, as workers in different kinds of institutions – hospitals, healthcare facilities, educational institutions, commercial kitchens and restaurants. Commercial services, more than privately employed servants, replace women’s unpaid labour in the vast majority of western families.
It’s impossible to create measures of hierarchies of emancipation across societies.
There are other solutions too. Migrant women too need to replace their own unpaid work in the home. In many developing countries people live in complex households where many women share the tasks of reproductive and care work. The decision of an individual women to work for wages away from the home has less negative impact in these contexts. That's because domestic tasks – childcare, cooking and cleaning, care for the sick – have been socialised in a different way, not through commodification and commercialisation, but through collective kinship responsibility for reproduction.
However – and I think this is a big issue – from my perspective as a feminist scholar, I believe it’s impossible to create measures of hierarchies of emancipation across societies. I would absolutely dispute any measure that claimed women in western societies are more emancipated than women in less developed societies: if you look at rates of education or political participation or even employment, many of the world’s developing countries have higher female representation than in the UK or Canada or the US.