You are correct about the vast time/information lag. Even with modern technology there still exists a lag of microseconds among different traders based on their geographic distance from the exchanges, and viable trading strategies have been designed to take advantage of this lag. You can imagine how much greater the lag was before the advent of modern telecommunications.

People who wanted to buy and sell shares used to meet in exchanges like the New York Stock Exchange. Here, they used the open outcry method, where people would use shouts and hand signals to indicate what they wanted. Brokers would record trades by hand onto papers, which they would then reconcile with their counterparties later on. The time lag would be mitigated as much as possible because all the major brokers knew to meet at the exchanges for trades, but of course there would still be delays.