Democratic organizations need leaders too – the main difference is that in a democratic organization leadership is dynamic and distributed. In a traditional organization leadership is static and centralized.
A common misconception is that democracy entails a lack of strong leadership, but this is only true of dysfunctional democracies. Healthy democracies have many leaders, who step up whenever interest, passion, energy, motivation and organizational needs dictate.
You may see different people taking leadership in different areas, or you may see people taking leadership in the same area at different times. But note that leadership in a democratic organization is something you actively seek out – whereas in a traditional organization it “comes with the job”.