I read a short story a dozen or so years ago in a short story collection with a yellow cover.
From your description it sounds like "Don't Look Now", a short story by Henry Kuttner (but see Addendum about authorship at the bottom of this post). Does any of these covers look familiar? The only yellow cover I see there is on the 1977 anthology The History of the Science Fiction Magazine, Vol. 3 1946-1955 (Michael Ashley, ed.) which would have been around for some time when you read it "a dozen or so years ago". The story is available at the Internet Archive as the original publication in the March 1948 Startling Stories, and as reprinted in the anthology My Best Science Fiction Story, and as a reading on the Mind Webs radio program.
The story was about a man - a detective I think (not sure, though) - who was sitting at a bar.
It's a conversation between two guys in a bar:
The man in the brown suit was looking at himself in the mirror behind the bar. The reflection seemed to interest him even more deeply than the drink between his hands. He was paying only perfunctory attention to Lyman's attempts at conversation. This had been going on for perhaps fifteen minutes before he finally lifted his glass and took a deep swallow.
One of them is a reporter:
"About the Martians. All this won't do us a bit of good if you don't listen. It may not anyway. The trick is to jump the gun—with proof. Convincing evidence. Nobody's ever been allowed to produce the evidence before. You are a reporter, aren't you?"
Holding his glass, the man in the brown suit nodded reluctantly.
The other seems to be an inventor:
"Well, I got my brain scrambled, in a way. I've been fooling around with supersonic detergents, trying to work out something marketable, you know. The gadget went wrong—from some standpoints. High-frequency waves, it was. They went through and through me. Should have been inaudible, but I could hear them, or rather—well, actually I could see them. That's what I mean about my brain being scrambled. And after that, I could see and hear the Martians. They've geared themselves so they work efficiently on ordinary brains, and mine isn't ordinary anymore. They can't hypnotize me, either. They can command me, but I needn't obey—now. I hope they don't suspect. Maybe they do. Yes, I guess they do."
He was starting to get restless about a possible alien infiltration within human society.
"Then you ought to be taking it all down on a piece of folded paper. I want everybody to know. The whole world. It's important. Terribly important. It explains everything. My life won't be safe unless I can pass along the information and make people believe it."
"Why won't your life be safe?"
"Because of the Martians, you fool. They own the world."
[. . . .]
"Wait," the man in brown objected. "Make sense, will you? They dress up in human skins and then sit around invisible?"
"Only now and then. The human skins are perfectly good imitations. Nobody can tell the difference. It's that third eye that gives them away. When they keep it closed, you'd never guess it was there. When they want to open it, they go invisible—like that. Fast. When I see somebody with a third eye, right in the middle of his forehead, I know he's a Martian and invisible, and I pretend not to notice him."
The particularity was that aliens could only be seen ("seen" might be a bit too strong, "distinguished" maybe) at the edge of one's vision. That is to say, one couldn't look directly at an alien, but only make out a presence almost at one's side.
The brown-suited man nodded. He took up the prints and returned them to his watchcase. "I thought so, too. Only until tonight I couldn't be sure. I'd never seen one—fully—as you have. It isn't so much a matter of what you call getting your brain scrambled with supersonics as it is of just knowing where to look. But I've been seeing part of them all my life, and so has everybody. It's that little suggestion of movement you never catch except just at the edge of your vision, just out of the corner of your eye. Something that's almost there—and when you look fully at it, there's nothing. These photographs showed me the way. It's not easy to learn, but it can be done. We're conditioned to look directly at a thing—the particular thing we want to see clearly, whatever that is. Perhaps the Martians gave us that conditioning. When we see a movement at the edge of our range of vision, it's almost irresistible not to look directly at it. So it vanishes."
I think the story began as the narrator entered the bar,
There's no narrator in the story, it's told in the third person.
progressed as he told someone his story, and ended when he himself sees something confirming his theory.
The story ends with the reader seeing something confirming the theory:
They shook hands firmly, facing each other in an endless second of final, decisive silence. Then the man in the brown suit turned abruptly and walked out of the bar.
Lyman sat there. Between two wrinkles in his forehead there was a stir and a flicker of lashes unfurling. The third eye opened slowly and looked after the man in brown.
Addendum about authorship. Henry Kuttner and his wife Catherine Lucille Moore (who wrote as C. L. Moore) famously collaborated on their writing. On Moore's wikipedia page we read:
Moore met Henry Kuttner, also a science fiction writer, in 1936 when he wrote her a fan letter under the impression that "C. L. Moore" was a man. They married in 1940 and thereafter wrote almost all of their stories in collaboration—under their own names and using the joint pseudonyms C. H. Liddell, Lawrence O'Donnell, and Lewis Padgett—most commonly the latter, a combination of their mothers' maiden names.
From Kuttner's Wikipedia page:
Kuttner was known for his literary prose and worked in close collaboration with his wife, C. L. Moore. They met through their association with the "Lovecraft Circle", a group of writers and fans who corresponded with H. P. Lovecraft. Their work together spanned the 1940s and 1950s and most of the work was credited to pseudonyms, mainly Lewis Padgett and Lawrence O'Donnell. Both freely admitted that one reason they worked so much together was because his page rate was higher than hers. In fact, several people have written or said that she wrote three stories which were published under his name. "Clash by Night" and The Portal in the Picture, also known as Beyond Earth's Gates, have both been alleged to have been written by her.
L. Sprague de Camp, who knew Kuttner and Moore well, has stated that their collaboration was so intensive that, after a story was completed, it was often impossible for either Kuttner or Moore to recall who had written which portions. According to de Camp, it was typical for either partner to break off from a story in mid-paragraph or even mid-sentence, with the latest page of the manuscript still in the typewriter. The other spouse would routinely continue the story where the first had left off. They alternated in this manner as many times as necessary until the story was finished.
As for the story "Don't Look Now", the ISFDB credits it solely to Kuttner. However, in the 1949 anthology My Best Science Fiction Story (Leo Margulies and Oscar J. Friend, eds.), Kuttner concludes his introduction to this story (available at the Internet Archive) with the line:
Anyway, my wife wrote it.
It seems probable, then, that C. L. Moore wrote part or all of the story "Don't Look Now".