There is a theory of money that says that the value of private bank money derives from the need of debtors to obtain money in order to settle their debts. Some people see this as being rather circular. If the value arises from the existence of debts, how do those debts come about? How can you borrow or lend something without knowing its value in the first place?
This is where it is useful to see that debts created through bank lending do something special, that other debts on their own do not. A useful way to see this is by looking at the monetary circuit described by circuit theorists. In this scheme, banks lend to firms to enable them to pay their workers. The bank loans create the money, which the firms then hand over as wages. At a later stage, workers spend their money, buying goods from the firms. The firms use the money they receive as sale proceeds to repay their loans.
It is assumed here that firms need to pay their workers in advance. Production takes time, so the firms cannot simply exchange goods for labour. One possibility would be for firms to pay workers with contracts for delivery of goods at a future specified time. So, for example, a worker might receive coupons entitling him to receive 100 widgets at the end of the month. This is fine, but if the worker doesn't want widgets himself, he then has to find people to exchange his widgets (or widget coupons) with, in return for the goods he actually wants.
On the face of it, money does not help here. For at least the widget coupons entitle the holder to a specified amount of something. Money doesn't entitle the holder to anything. If firms just printed up some coupons, with no specific entitlement attached to them, and tried to use these to pay their workers, the workers would quite reasonably reject them. They would have no reason to believe that anyone would subsequently accept them in return for anything of value.
Involving banks means that the coupons can become tokens for the cancellation of debt. Each firm can (in fact must) use coupons to repay their loans from the banks. The coupons thus have value, without the need to be linked to a specific commodity. Firms need to obtain coupons to meet their contractual obligations to banks. Workers will accept coupons as wages, because they know firms will be prepared to accept them for goods at a later stage.
So creating value in the coupons specifically requires the involvement of the bank. In the monetary circuit, the firms are the debtors, the workers the creditors and the banks are just intermediaries. But although they only intermediate, the role of banks is critical to establishing a money denominated in an abstract unit. Firms and workers can create contracts through bilateral agreements, but these can only ever be linked to specific commodities. Credit money requires a tripartite arrangement.
Of course, there is still no guarantee for holders of coupons of what they will be worth in the future. They know there will be a demand for coupons, but if there is a rush of new lending there may be an excess supply of coupons depressing the value. People's belief in the intention of banks (specifically the central bank) to control the rate of credit creation is crucial to establishing the value of this money.