Is that they are basically like modern day checks except that the banknotes aren't written to any specific person and can be used like currency.
They got cleared like modern day checks do though. If one bank gets banknotes from another bank they settle the debt as if it were a modern day personal check written by someone who was a customer of another bank.
Make sense now?
So without a central bank, if a bank issues notes too promiscuously then they will get killed by clearing debt. Just like as if someone had written a bunch of bad checks more or less... This is what keeps banks from inflating the supply of banknotes beyond what is prudent in a fractional reserve free banking system.
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