The screaming was not only during handshake. The acoustic coupler you see in the image was making that "scream" all the time (that's how the data was transmitted), so no noise, no communication/data.

People got so used to the noise that some of them could even diagnose connection issues based on the sound the coupler made.

When modems were updated they internally converted the data directly to electrical signals to be sent over the telephone via, without the audio indirection, but most modems had a built in speaker anyway that still played back the audio during handshake (on default settings) to allow users to hear/debug connection issues.

Practically all modems from that time allowed you to turn off that noise - even during handshake.

Over the years the need to "hear" the data went away.