Taxes were collected by Sheriffs in the western territories. When Tombstone became a major mining town in 1880/1881, the person responsible was the infamous Sherriff Behan of OK Corral fame.

County sheriffs wielded considerable power in their areas and were personally charged with carrying out their numerous duties from law enforcement, fee collection, taxation, and jailing. They were (eventually) paid substantial salaries to reflect this, but in the early 1880s would primarily take income from administrative fees for services rendered like serving warrants ($1.00-$2.00), calling witnesses ($0.05), and executing death warrants ($15.00) and a percentage of tax collection (~5%). These numbers are from the Kearny Code and would be substantially increased only a few years later in 1887 to reflect increasing costs in new counties with smaller tax bases.

Records indicate that Sheriff Behan was a fairly wealthy man by the standards of the time and area with an income of almost $25,000 (~$750k today), but mentioned previously sheriffs were in many cases personally responsible for the discharge of their duties. This in turn typically required them to personally hire their deputies and tax collectors, who would be paid out of the sheriff's income. They would also be expected to post substantial bonds ($10,000 each for the duties of law enforcement and tax collection) to the county as an insurance against abuse of office. We know how well that worked out with Behan, but it did exist.

As for how taxes were determined, the specific rules around this were largely up to the territorial legislature of the territory in question. In actual practice, the enforcement of these laws could be uneven at best. Deputy Assessors would often rely on landowners' assessments of their own properties or livestock counts to assess taxes. Ranches bordering a county or state line would avoid taxes more subtly by moving some of their livestock to conveniently seasonal grazing lands as appropriate.

In any case, the main deputy assessors for during the heyday of Tombstone were Wyatt Earp and William Breakenridge. Their records and most of the remaining primary documentation for Tombstone can be found at the Cochise County Archives in modern Bisbee, AZ. Additional records are also maintained by the Arizona Historical Foundation in Phoenix and Tucson.

[1] https://doi.org/10.2307/41859684

[2] Ball, Larry D. Desert Lawmen: The High Sheriffs of New Mexico and Arizona Territories, 1846-1912. UNM Press, 1996.