We had had transistor radios for some time, so the replacement of that by the Walkman was not particularly noticeable in the street. The concept of private listening had already been established.
Stereo HiFi personal music was a new thing though. Transistor radios were mono and were listened to through a low fidelity crystal earphone in one ear. The time was ripe for mobile personal cassette players though, business had been using small portable cassette recorder/players for a while to dictate notes while away from the office. They became very cheap; I bought one in college and connected it to the Sennheiser headphones I used with my stereo using a hacked adapter from Radio Shack. It was mono but I had mobile cassette music perhaps a year before the Walkman came out. It was kind of surreal to have my own personal music while walking, where I could listen to lots of stuff that would never get radio play. For example, listening to King Crimson while walking through a department store. It made the world seem different, I had more control over the environment, I didn't have to listen to the Musak or hear people talking. People would stop me and ask about the system and go off and build their own. There were probably quite a few people who invented their own before Sony released the Walkman.