Someone once told me, back in my supermarket working days of the nineties, “right over left that’s men’s left over right that’s ladies sir, now get back to folding those tables”. And that was my enlightenment on the subject for fifteen years. It’s fine. It’s not wrong. But it’s trivia. It gets someone else to the answer on a pub quiz question and ignores the real reason, which is that the fact that buttons are on one side or the other tells you something concrete about how the coat is cut, and if you find yourself asking why there’s usually a reason you’re asking.
You’ve already noticed something.

Either your own coat doesn’t hang right when it’s done up, or you’ve tried one on in the changing room and it’s sitting oddly across the chest and don’t know why. It’s not trivia. Button side is a symptom of something.
Why it’s right and why that should still matter to you
The story I hear quoted most often these days is the sword story, the need for men to have their right hand free to draw across their body while keeping their coat under wrap. Frankly I’ve never bought it completely. Mostly because gentlemen who’d never seen a sword were still dressing the same way hundreds of years later while tailors who’d never drawn a weapon in anger were still teaching apprentices how to coatbind. The theory I find more useful, and importantly the one that actually impacts on how they fit, is simpler. Men traditionally dressed themselves. Buttoning right across the body is easier for a right handed person to do themselves rapidly, thumb and fingers pulling the button through the hole without crossing over any other part of the hand. Women for most of the years this convention was established had clothes put on them more often than not, by someone else, usually a maid. And if you’re dressing someone else it’s easier to button them up left across right if you’re facing them. Believe that servant explanation or not, what you’re left with now is decades of the single method surviving well past the reason it was originally chosen, and it’s a difference now built into how they’re cut as well as which way the flap lies.
This is the part Malcolm didn’t explain to me. When you start with a coat that closes right across left the entire garment flows from that, the way the pieces overlap, where the inner facing is positioned, even where the top button sits relative to the chest, all of it assumes you have a certain sort of shape inside those sleeves. Broad shoulders, straight torso, no breasts. Dress a coat that buttons the opposite way round and there are darts and shaping inside it designed for a very different sort of shape. So the button side isn’t aesthetic, it’s not putting some decoration on what is essentially the same garment underneath. Button sides mark coats that are constructed on entirely different patterns from each other, and that’s the important thing when you’re trying them on, not which way will make you look right in the mirror.
The thing this actually tells you when you’re stuck there in the changing room
Let me say something which I think will sound incredibly obvious now I mention it but obviously isn’t, because this question bafflingly gets asked more than anything else I cover. If you’re standing in front of a coat, it has the buttons on the left and you’ve got a man’s body. No variation of sizing is going to make that sit right on you. Try the larger size and it will still feel like it’s pulling across the chest, because the issue isn’t with the number on the label, it’s with where that number was applied. Go big or go home, it makes no difference either way. I did that very thing myself once with a wool overcoat I picked up from a vintage stall roughly a decade and a half ago. Lovely colour. Super heavy stuff. The kind of coat you buy because it looks perfect hanging up and rather less think about how it’ll actually fit when you put it on. Got it home, couldn’t for the life of me figure out why every time I did up it looked wrong.

Had the most irritatingly long stroll around my flat with it hanging off one arm before I realised the buttons were on the wrong side for a men’s coat, it was a women’s coat from the start and I’d never noticed while I was buying it cause I was too busy looking at how lovely the wool was.
The person searching that on google, in myexperience has likely already figured half of that out already.
They’ve noticed their coat hangs weird compared to the rest of their wardrobe. Or they’re standing in Tesco having to hold two coats up so they don’t fall down, both of which look identical except for this one thing and they want to know if it’s actually important or if they’re just being picky. It is important. Not because there’s going to be a Witch Doctor at the door of the pub to turn you away if you get it wrong, although really don’t be that guy, but because knowing which side the buttons go on is the single quickest way to figuring out if a coat was likely cut with your shape in mind or someone else’s. My brother once tried on a coat in a charity shop, loved it, would not understand what I was saying about it, and pointing at the buttons and saying nothing else was enough for him. Walked straight back to the pile.
If you take anything away from this let it be this; look at the buttons long before you look at the size label. The label will tell you a number that someone else thought would fit you. The buttons will tell you which body that number was intended to fit around.