From what I can tell (this is not my field), at least as concerns the English state, the tally stick was divided into the "stock," which was kept by the state, and the "foil," which went with the individual.
This video shows a number of examples of stocks (apparently few foils survive) and how they recorded debts. https://www.youtube.com/watch?v=0rxAYSXJ41Y .
It explains that they were used to record money that was deposited, essentially a receipt, by removing wood from the then original stick before it was split into the stock/foil. My guess (again not my area) is that it primarily functioned this way:
1. The removed wood representing the debt. I have no confirmation, but I would expect the stock/foil was created in view of the depositor who understood its markings.
2. The entity seeking repayment would not be able to fake their receipt, presumably to indicate they had deposited more money than they had (i.e., by removing wood from their foil). If they tried, the stock would still have that wood present, but the other grooves and grain of the stock/foil would align. The state would easily be able to tell that someone had subtracted wood from their foil.
3. The state could, of course, destroy the stock and claim its nonexistence, but information of that would, presumably, escape and others would be less likely to lend to the state. It appears there is writing on the stock/foil, so perhaps that served a secondary verification function (e.g., handwriting comparison or reference to a paper ledger)?
[EDIT: The process of making English tallies was laid out in the "The Dialogue Concerning the Exchequer. circa 1180" (https://avalon.law.yale.edu/medieval/excheq.asp. It describes making these tally sticks in the section "As to the Maker of Tallies." Unfortunately, I do not see the details of dispute resolution in that document. But, I only skimmed it.]