Is zakat due on money borrowed for two years, to be paid by the borrower or its owner?

You are not obligated to pay zakat (definition) on borrowed money. Rather, its zakat is incumbent upon its owner, who lent it to you, provided that a year has passed over it, it has reached the zakat threshold (nisab (definition)), and the debtor is solvent and not procrastinating. If the debtor is in hardship or procrastinating, its zakat is not obligatory upon its owner until he reclaims it, after which he pays zakat for one year.