Is it obligatory to return the deducted amount to the person whose debt I assumed after settling the debts with their claimants, knowing that some of them waived a portion of their debt?

What you both agreed upon is an agency and a loan. Your friend borrowed money from you and appointed you as an agent to pay off the debt on his behalf. You should only claim from your friend what you have paid. Whatever the creditor waives from the due debt is dropped, and your friend will not be asked for it.

Waiving part of a due debt is permissible, unlike a deferred debt. Waiving part of a deferred debt in exchange for accelerating the payment of the remainder is not permissible, because it constitutes usury (riba).