I understand that the distinctive way the two pieces of a tally stick and the markings on them fit together prevented altering records. But that system only works if you can bring the pieces of a tally stick back together. If you were in debt, and the only way a third party could verify that was with your half of a tally stick, why not just get rid of the thing and claim that you never borrowed from your creditor? You could keep a record somewhere else, but surely that defeats the purpose of the tally stick. You could be adversely affected by reputational damage in the long run, but again the tally stick isn't doing anything. Your reputation is inducing you to pay the debt.
The tally stick is a document of a transaction just like any other. It doesn't 'do' anything any more than a land deed 'does' something or a modern credit card statement 'does' anything. It all depends on the trust given to the document by others. All documents depend on the collective faith in those documents, and the existence of agents (whether they be bailiffs, police officers, sheriffs, lawyers, whatever) who will enforce the claims made in the documents if need be. Somebody could conveniently lose their half of a tally stick, but the other party still had evidence that the loan took place, and could ask for redress.
Thus far the split tally stick was just a document like any other. However, the design of the split stick included inherent additional security to both parties over a simple 'receipt' or 'statement' in the modern style, because its design mitigated any attempt by the debtor to claim they owed a smaller amount, or the creditor to claim they loaned more than they actually did. In other words, the stick was in most important regards tamper-proof. That didn't take away the need for society at large to trust that the stick was not forged for some reason by both the parties, and it didn't prevent somebody trying to claim they had no knowledge of the loan. But in that latter case if a creditor held half a stick and was viewed as trustworthy by the local authorities, the person who denied the existence of their half of a stick was going to be on legally shaky ground, just as they would be if they denied the existence of any other document that was still in the possession of another person.
Further to MCW's comment, I did a little bit of searching for some examples of tally sticks being used as evidence of debts or payments. It doesn't exactly show an incidence of somebody denying a tally exists, but does show how the existence of a tally (or one side of tally in the hands of a debtor or creditor) was viewed and respected by authorities. The date of the law regarding sheriffs is not recorded in the Register of Malmesbury, but is before 1400.
Example of tally stick being used to seek redress of unpaid debt
(Acts of the Privy Council of England. Volume 42: 1627, January - August), p. 59. Extracted from https://sources.tannerritchie.com/browser.php?ipid=694108)
Law regarding illegal debt collections by sheriffs
Registrum Malmesburiense. The Register of Malmesbury Abbey, preserved in the Public Record Office. Volume I [Rolls Series]), p. 508. Extracted from https://sources.tannerritchie.com/browser.php?ipid=1356302
@fred2's answer is a good one and I've upvoted it, but it's also worth remembering the societies in which these people lived and made their transactions. They were small, local societies where everyone knew everyone else. The fact of a transaction would usually be well-known and a matter of village gossip. Even in 19th century America -- where communication and movement was much, much greater and the local society larger than in a medieval or ancient society -- who owned what and who borrowed what from whom was one of the few topics possible for conversation.
"Well, Joe McGinnis's dad bought the land north of Hay Creek up to that hill top over there from Pete's grandfather maybe fifteen years ago -- remember, his barn burned down and he needed the cash to rebuild it? -- and Joe's dad had always wanted that hillside. I heard that Joe's dad had to give Pete's grandfather three head of cattle as well as money."
The fact of a transaction would be well-known. (It was very much in both parties' interest for the fact of a transaction to be well-known in a society which did not yet have a local Registry of Deeds, since for a property dispute to be settled in court, it would be done by examining any deeds and the memories of witnesses.)
The same applies to a lesser extent for any kind of transaction: The ability of a court to settle who owes what and who owns what relies on the memories and testimony of the community.
But while the fact of a transaction would be well-known, the details of subsequent payment would typically be less so, since they would be smaller transactions -- e.g., annual payments of a fraction of the amount owed -- and less interesting and less likely to be remembered. And there's where the tally sticks come in.
For Joe (or Pete) to destroy his one of the tally sticks would be futile, since everyone knew he'd had it and his family had bought the land and if he could not produce it, he would have to accept whatever the other claimed.
Your tally stick protected you from any unjust claims of the other party. There's no reason to destroy one.
The modern equivalent of a tally stick would be the two copies of a contract kept by each side in the transaction: the point of having a copy is having the evidence that the transaction did happen and that the other side agreed to the terms stated in the contract.
Thus, destroying one's own copy copy forfeits the one's ability to prove that the transaction took place or to prove the one's claims regarding the conditions agreed upon. Obviously, having the evidence does not guarantee that one can enforce the contract - this requires external enforcement mechanisms, like courts, reputation, or public knowledge or simply good faith.
Another answer that relies on logic rather than evidence.
At least to my read of Wikipedia:TallyStick, tally sticks recorded partial payment of taxes. (There is no need for an artifact to substantiate that you have paid none of your tax)
Destroying a record of partial payment would not benefit the taxpayer at all, and the government has other mechanisms to undermine the contract.
I'd be interested in records of court cases that covered challenges to tally stick contracts.
T. Anton points out that tally sticks have a longer history than the Wikipedia article indicates. I believe this to be true, but I don't have the time to support that belief with evidence. (if I had better scholarship, it would require less time).