I don't have a thorough answer about the mechanism of substitutive atonement -- in general, in Judaism, a person dies for his own sin. In those instances where it LOOKS like one is dying for another, the rabbis explain that the death of a particular person is designed to inspire others to repent and avoid their own punishment. The death brings about atonement by driving a change in the other person or people.
However there are a few other important concepts which I am willing to piece together:
- [I don't have the medrashic source for this]
"Chazal tell us that if the nations knew how much the Bais HaMikdash
benefited them they would not have destroyed it. They would have
placed armed guards around it to protect it. "
So Israel's prayers and sacrifices (and one might assume, the continued prayers after the sacrificial system ended) had an effect on behalf of the world which would, I guess, include atonement through prayer.
- From www.drazin.com
Yet, amid all this persecution, the Jews have relentlessly "interceded for the [Gentile] transgressors," in compliance with:
And seek the welfare of the city to which I have exiled you, and pray to the L-rd on its behalf, for in its welfare you will have peace.
So despite suffering at the hands of the nations, by following the injunction to pray for the well being of the various governments, Israel has achieved results on behalf of others.
- The Drazin site also includes the following (though without a source for the idea):
The Jews have certainly borne the iniquities of the Gentiles: throughout history, they have been discriminated against, oppressed, tortured, raped, robbed, and murdered. By submitting themselves to this treatment and acknowledging it to be His will, they have justified and sanctified G-d's Name.
This goes back, somewhat, to the statement I made above -- one person/group's suffering should inspire others by showing the suffering/death as a statement of piety which drives others to try and be more pious.