The code for the website, backoffice systems, credit scoring, and fraud detection could be open-sourced and continually improved.
Data on credit decisions would be available to any member to scrutinize and challenge, thus reducing fraudulent or unwise loans and identifying opportunities to expand profitable lending.
The bank's individual loans and investments would be public, perhaps lightly scrubbed of personal identifying information. Regulators, journalists, and motivated individuals could see if sweetheart deals are being made at the expense of the membership or the deposit insurance fund.
The bank could publish a public "blockchain" of transactions similar to Bitcoin so financial statements could be reconciled down to the transaction level. This might make it much more difficult to hide fraud and abuse.
Incidents of fraud and default would be published as they occur. The resulting social pressure should reduce the bank's costs, and the data could be used to improve credit and membership decisions.