eu-cookie-law does work in local WP installation but not in installat on the Web

When installing, activating and configuring the eu-cookie-law plugin in my local installation for up-front test it correctly shows the banner upon cache cleaned Browser on mutliple devices linking to my local net.
But when I install the same version (3.0.6) in my public website, configure it the same way, the banner is not visible to any other user, only to my own PC as long as I am logged in to the WordPress of my Web installation which is hosted with a hosting company.
What can the reason be?

Waiting for response urgently, Joerg