In a free market, legitimate contracts cannot violate private property rights. Freedom of contract is contingent on there being no violation of private property rights. For example, a contact between a mobster and a wife to murder her husband for $10,000 is illegitimate and would not be defended by the law in a free market. Even the advocates of fractional-reserve, free banking don’t claim that the current treatment of fractional-reserve accounts would be legitimate on the free market. This is because fractional-reserve accounts give ownership claims to their cash reserves to (roughly) 10 different claimants at the same time. For this reason, your time-share example is dis-analogous. The analogous situation would be that for each unit a time-share company had it gave instantaneous claims to 4 different persons.
Proponents of fractional-reserve free banking, then, assert that in the free market banks would offer checking accounts that were lottery claims to the cash reserves. While such contracts might be legitimate on the free market (unlike the simultaneous, instantaneous claims of our current system), it’s highly unlikely that merchants would accept such account balances as a medium of exchange. After all, merchants can insist on being paid in money instead. Although banks could pay their customers interest as incentive to hold fractional-reserve accounts, it’s difficult to see how they could give incentive to non-customers (i.e., merchants) to accept the checking account transfers of their customers.
Here are some readings:
In summary, my view is that in the free market banks would be free to make any legitimate contract they desire with their customers, including contracts for lottery-ticket style accounts or near-instantaneous- loan accounts. But such accounts would not become money substitutes and therefore, would not be part of the money stock. the money stock in a free market would be commodity money and money certificates. By restricting the money stock to money and money certificates, monetary inflation and credit expansion are eliminated and with them, business cycles.