Was handshake common among the Companions of the Messenger of Allah (may Allah's peace and blessings be upon him)? He replied: Yes

Qatādah asked if the Companions of the Messenger of Allah used to shake hands when they meet and great each other with Salām as an extra symbol of love and honor. Shaking hands is done with the right hands and causes the sins of those doing it to be forgiven before they separate and depart. This shows the virtue of handshake upon meeting someone to talk with him or the like. However, mere meeting in the marketplace does not require handshake for this was not the practice of the Companions.

In other words, if you come across people in the marketplace, it is enough to greet them with Salām only.