Mechanically speaking, there are two functions that have to be centralised, although the second one need not be performed by the central bank historically it usually is.
There is an absolute requirement for a guaranteed lender of last resort, since banks can run into liquidity issues through no fault of their own. Liquidity in this context, is the availability of asset cash or equivalents to meet intra-bank transfers (say money is transferred between deposit holders at two different banks.) A bank can be solvent long term, i.e. their loan book is healthy so they are guaranteed to receive cash long term, but still have issues in the short term - if a big transfer occurs. (Inter-bank lending is necessary but not sufficient for sorting out short term liquidity issues because of competitive issues.)
The second issue is how the banking system's expansion of the money supply is regulated. Historically reserve accounts at the central bank were used for this, and that mechanism can provide absolute control, provided it is applied to all liability deposit accounts in the system. Other mechanisms could be designed that didn't use a centralised method - indeed many banking systems currently no longer use reserve accounts, and are relying on a mixture of Basel capital controls, and control over other factors, such as direct limits on borrowing - the long term effectiveness of these controls is yet to be demonstrated though.
A third issue is who is going to handle physical clearing, generally these days that's not done by the central bank, but occasionally small countries do it that way for expediency (using the reserve accounts).
Apropos the old gold standard. Gold was treated as an asset in the system, and effectively regulated lending, and consequent deposit expansion. However, it wasn't a very good regulator in and of itself, since there was a feedback relationship embodied in the price of gold, which was affected not only by the banknotes issued by the banks, but also by deposit expansion. Every time somebody wrote a cheque, and bought gold with it, the quantity of bank deposits also impacted its price. (Leaving aside a whole bunch of other instabilities surrounding the quantity of bank notes being issued by private institutions.)