I think the problem with "someone was staring at me" sixth senses is that you only need to be right once to think that it was real. I've felt that I was being stared at loads of times. To the best of my knowledge, it's never been true, really. Certainly not often enough to be a real sense. More likely to be just my rampant paranoia again.
I was due to meet someone for coffee at about 2pm and thought that they were going to cancel about 90 minutes before. They then asked that it be moved to 15 minutes later, but at about 12.20pm they seemed happy to go. Then, with about 90 minutes left until we were due to meet, they sent a cancellation text. Spooky? Not really, I usually fear the worst.
In another, more tragic case, on the day when I found that all three of my pet hens had been killed (by a fox, most likely), I spent the entire journey from my back door, where I could not see them, to the entrance to their coop, knowing instinctively that they had to be dead or missing. No sixth sense again -- just a sadly correct logical deduction based on the fact that there were no hens coming to greet me at the coop entrance. It made for the worst journey of my life.
I don't know if either of the above cases is truly relevant, but I suppose the main point I want to make is that if you expect something, and it later turns out to be true, that doesn't imply anything because such incidents are often isolated, and only memorable because of the fact that the prediction turned out to match the outcome. How often does it not?
Still, we risk turning this into yet another argument. At the moment it's clear that we don't know even close to everything about how the brain and the current senses work. As long as that is the case, and as long as there is the simpler explanation of confirmation bias, we don't need a new "sixth sense" to explain these phenomena. It's more likely to be just a combination of chance and the fact that humans are notoriously bad at understanding how chance works.