The most quoted sentence in the rabbinic literature on atonement is a limitation rather than a promise. The tractate of the Mishnah devoted to the Day of Atonement states that offences between a person and God are atoned by the Day, and that offences between a person and their fellow are not atoned by the Day until the injured party has been appeased.
The consequence organises an entire season of Jewish practice. The days before Yom Kippur are the time for approaching people one has wronged, and the approach has to be made in person and has to be genuine. The liturgy of the day itself cannot substitute for it. Later authorities worked out the details with the seriousness the ruling invites: how many times a person must ask if they are refused, what is done if the injured party has died, and whether a general apology reaches a specific wrong.
Two things follow that bear directly on comparison with the Christian material.
The first is that atonement here is not a single transaction but a set of conditions, most of which are the offender's own work. Repentance is required, restitution is required where restitution is possible, and the Day supplies what those cannot. Nothing in the sequence is done by anyone in the offender's place.
The second is that the same tractate is candid about what remains hard. It preserves a discussion in which different classes of offence are said to be met by different combinations of repentance, the Day, suffering and death, which is not the arithmetic of a mechanism but the working out of a case law. Read alongside the Christian theories, the difference in genre is as striking as any difference in content: one tradition is arguing about how the thing works and the other is arguing about what a person has to do.