It would probably be at least somewhat determined by what you're physically close to, and what a person's capabilities and interests are (one that opted for medical training could be rotated into needed medical roles), plus what they're personally willing to do.

we’re still in the position of a dystopian “today it’s your turn to work in the salt mines, comrade” decision being made by a higher power, and imposed by law.

There would be no state government in a libertarian anarchist society, and no institution that could force anyone to do anything against their will (if anything like that began to form, it would be in society's best interest to dismantle it).

Ultimately the 2 to 3 months of work would have to be entirely voluntary, otherwise it would be dystopia.

You may then think that the whole concept comes crashing down in a house of cards, as there's simply no way anyone would engage in that essential work without some coercive force above them making them do it, and thus everything becomes unmaintained, and everyone tries to free-ride on it until the the whole system collapses on itself. But this issue has been pondered for over 100 years now, and there are compelling counter-arguments that it's still extremely viable.

As an example, this aspect was discussed at length by Peter Kropotkin in The Conquests of Bread (specifically Chapter 12: Objections).

It also breaks down why the Authoritarian Marxist argument of forcing people to work will only lead to the same conditions we're trying to escape from:

“But the danger,” they say, “will come from that minority of loafers who will not work, and will not have regular habits in spite of excellent conditions that make work pleasant. To-day the prospect of hunger compels the most refractory to move along with the others. The one who does not arrive in time is dismissed. But a black sheep suffices to contaminate the whole flock, and two or three sluggish or refractory workmen lead the others astray and bring a spirit of disorder and rebellion into the workshop that makes work impossible; so that in the end we shall have to return to a system of compulsion that forces the ringleaders back into the ranks. And is not the system of wages paid in proportion to work performed, the only one that enables compulsion to be employed, without hurting the feelings of the worker? Because all other means would imply the continual intervention of an authority that would be repugnant to free men.” This, we believe, is the objection fairly stated.

It belongs to the category of arguments which try to justify the State, the Penal Law, the Judge, and the Gaoler.

“As there are people, a feeble minority, who will not submit to social customs,” the authoritarians say, “we must maintain magistrates, tribunals and prisons, although these institutions become a source of new evils of all kinds.”

Therefore we can only repeat what we have so often said concerning authority in general: “To avoid a possible evil you have recourse to means which in themselves are a greater evil, and become the source of those same abuses that you wish to remedy. For do not forget that it is wagedom, the impossibility of living otherwise than by selling your labour, which has created the present Capitalist system, whose vices you begin to recognize.” Let us also remark that this authoritarian way of reasoning is but a justification of what is wrong in the present system. Wagedom was not instituted to remove the disadvantages of Communism; its origin, like that of the State and private ownership, is to be found elsewhere. It is born of slavery and serfdom imposed by force, and only wears a more modern garb. Thus the argument in favour of wagedom is as valueless as those by which they seek to apologize for private property and the State.