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How can I activate the Widget

As soon as you activate this plugin go to Widgets under Appearance, find "Network Login" and drag&drop it onto your sidebar.

I've activated the Widget and my sidebar is different

This is normal if you didn't use the Widgets in the first place. As soon as you add a Widget you'll be using the dynamic sidebar, therefore you need to configure it by dragging all the content you want from the left side onto the sidebar.