The authorities holding the sticks are acting as a public ledger, in that
they bear witness to person A originally handing them half of stick A — and they could write that association down, if they like (picture a little tag saying "person A" wrapped around stick A in their stick storage closet); but even if they don't write it down, they may just remember the person, remember the probably-pretty-unique-looking stick, and mentally associate the two.
Then, when person B comes around with the other half of stick A, they won't know all the hands it passed through in the meantime, but they'll at least know that person A and person B were connected through some chain of equal-value trades that didn't require "breaking" the stick.