How can I make a metal privacy screen more private?
Your first approach of random color spots would work, but as you suggest, it might not look pleasing to your eyes. The other suggestion you have about painting a nicer yard on it is "spot on."

Anything you do to the mesh to break up the pattern currently in place will serve to hide that which shows through the mesh. Abstracts and landscapes and other images that have multiple shapes will work better than geometric patterns. 

The military did some studies regarding camouflage painting and found that very little variety worked quite well, as long as it was irregular or scattered. 

You could duplicate the block-work adjacent to the gate, but it has such a uniform appearance that it likely would fail to hide that which can be seen through the mesh.

The real answer is something for viewing that meets your opinion of attactive, while maintaining a broken or segmented image.