Under system where people can hold central bank (CB) created e-money private bank issuance of demand deposits self regulates. Assuming CB e-money is equally as functional for payments people will mainly hold demand deposits (zero interest money) at the central bank because it is risk free whereas commercial banks are not. Private banks can create money but unless it pays interest people will shift money away into risk free central bank accounts which will drain banks of assets. Therefore private bank created money will carry an interest cost to issuers and this will regulate the amount of issuance. Banks will issue money in accordance to the returns they receive from proceeds of invested money.
Banks that make highest returns from investments will attract most depositors because they can offer higher returns on deposits or other savings vehicles/investments they offer to customers.
The most transparent and least risky banks will attract most depositors.