Nowadays, the underlying instrument (i.e. a fancy piece of paper) rarely ever changes hands. It's possible to get it (with some delay) but most people don't bother.
The digital/electronic record of the trade and the transfer of monetary funds between accounts is generally sufficient to establish ownership and both happen in near real time under SEC regulations between registered brokers.
Basically, if the cross checked double entry accounts of buyer, seller and their brokers all agree that the trade took place, that is good enough for most folks and the SEC. Day traders can and often do buy and sell a stock multiple times in a day without delay.