The effect is the same whether it's one bank acting alone or many banks acting in concert. One dollar deposited at one bank leads to a series of loans and further deposits which eventually permit 1000 banks to make 1000 $1 loans and, all together, collect $10 worth of interest. Provided, of course, that no two individuals ever ask to withdraw "their" dollars at the same time and thus bring the whole house of cards crashing down.
Out of curiosity, do the banks really need to be different, or does money deposited by a bank's loan customer count toward that bank's deposits? It was my impression that all deposits counted and there was no need for the money to pass through many distinct banks.
I'm not really as fanatical about fractional-reserve banking as some. I consider it shady and unsound but not inherently fraudulent so long as the bank's practices are public knowledge; caveat emptor. I would prefer a clear dividing line between demand accounts, which the bank must be prepared to pay out in full at any time (full reserves), and time accounts, which should be clearly described as investments, loans to the bank which, like any loan, may not be available for withdrawal on demand and could end in default, costing you some or all of your principle. Current practice is to mix the two types in varying proportions, from interest-bearing checking accounts to CDs guaranteed against default.