Most people assume that money has always been issued by national governments but this is not so. Before the civil war the people used bank notes as money:. Here is an interesting excerpt form Wikipedia on the subject:
During 1861, the opening year of the American Civil War, the expenses incurred by the Union Government far outstripped its limited revenues from taxation, and borrowing was the main vehicle for financing the war. The Act of July 17, 1861 authorized Secretary of the Treasury Salmon P. Chase to raise money via the issuance of $50,000,000 in Treasury Notes payable on demand. These Demand Notes were paid out to creditors directly and used to meet the payroll of soldiers in the field. While issued within the legal framework of Treasury Note Debt, the Demand Notes were intended to circulate as currency and were of the same size as and, in appearance, closely resembled banknotes. In December 1861, economic conditions deteriorated and a suspension of specie payment led the government to cease redeeming the Demand Notes in coin. Also, in 1861, at General Grant's Headquarters, Edmund Dick Taylor mentioned his idea for greenbacks.The Legal Tender Acts
The beginning of 1862 found the Union's expenses mounting, and the government was having trouble funding the escalating war. U.S. Demand Notes—which were used, among other things, to pay Union soldiers—were unredeemable, and the value of the notes began to deteriorate. On January 16, 1862, in a private meeting with President Lincoln, Edmund D. Taylor advised him to issue greenbacks. Congressman and Buffalo banker Elbridge G. Spaulding prepared a bill, based on the Free Banking Law of New York, that eventually became the National Banking Act of 1863. Recognizing, however, that his proposal would take many months to pass Congress, in early February Spaulding introduced another bill to permit the U.S. Treasury to issue $150 million in notes as legal tender. This caused tremendous controversy in Congress, as hitherto the Constitution had been interpreted as not granting the government the power to issue a paper currency. "The bill before us is a war measure, a measure of necessity, and not of choice," Spaulding argued before the House, adding, "These are extraordinary times, and extraordinary measures must be resorted to in order to save our Government, and preserve our nationality." Spaulding justified the action as a "necessary means of carrying into execution the powers granted in the Constitution 'to raise and support armies,' and 'to provide and maintain a navy.'” Despite strong opposition, on February 25, 1862, President Lincoln signed the First Legal Tender Act into law, authorizing the issuance of United States Notes as a legal tender—the paper currency soon to be known as "greenbacks." In his correspondence, Lincoln credited Edmund Dick Taylor for his suggestion of the greenback currency, and named him "Father of the Greenback."Too much money will produce inflation too little money and you have a lack of demand or unemployment. In a free economy with fee banks issuing their own currency, banks are motivated not to issue too much currency because they could fall off par with the negatives that come with that. That are motivated to avoid deflation because that would represent lost opportunity to make profits.
Depositors and investors would be motivated to watch their banks but it would be rare for a bank failure to cause a total loss to depositors. Since people trust companies less than they trust Government, people would not be inclined to horde bank issued money in a downturn but if they did and the money started to get more valuable the issuer would be motivated to buy more assets and make more loans so as profit from its rising currency. This would increase demand when it is needed.
In such a system the failure of a bank would strengthen all the remaining banks and it would make it impossible for all banks to go down at once.