Contextual links for nodes displayed via views
Solved it myself.

Steps:


Enable Global: Contextual Links and Content: Edit Link fields
Exclude Content: Edit Link from display
In Global: Contextual Links settings, select the Content: Edit Link


However this works correctly and fine only while in preview mode, when watching the actual page in action the links nor the cog icon does not appear. As I understand this is some kind of a bug.

To fix that, you need to add additional CSS to your theme, which is (in my case):

.views-field-contextual-links {
  position: relative;
}


And now everything works fine, every element has it's own drop down cog/link.