But do they need to pay each other?

The exchange makes commissions in bitcoins from trades. It can sell bitcoins if it needs fiat money for operational costs.

Provided the exchange trusts the payment processors, it can allow their exchange balance to go negative (as would be needed if the processor has handled more deposits than withdrawals). In the long run, deposits and withdrawals should roughly balance (hopefully!).

If a payment processor's balance with the exchange becomes too negative, they could deposit funds via another payment processor. And similarly if their balance with the exchange becomes too high - they can withdraw via another payment processor. An agreement could be reached where payment processors do charge no or very low commission to other processors. This would also be necessary if a payment processor wants to cease processing payments.