Q. Which is correct: “one should do one’s duty” or “one should do his or her duty”—or, using singular they, “one should do their duty”?
A. In your example, one is closer to the personal pronoun you than to the indefinite pronoun everyone. Everyone would normally pair with his, her, or singular their, as in everyone should do their duty (see also CMOS 5.51). One, by contrast, can simply switch to the possessive case like other such pronouns:
I should do my duty; you should do your duty; he should do his duty; she should do her duty; they should do their duty; we should do our duty; one should do one’s duty
According to Bryan Garner, however, writers have tended to pair one with he (and, by extension, one with his), despite objections from “strict grammarians” and others (see Garner’s Modern English Usage, 5th ed. [Oxford, 2022], under “One . . . he”).
We can only hope, then, that we’re doing our duty as arbiters of style by recommending a pairing of one with one’s.