Both anecdotes are wrong, but both stem from an urban legend about the Japanese Arisaka rifle (WW2) which allegedly was deliberately chambered in .31 caliber so it could fire American .30 caliber munitions but the Japanese cartridge would not fit an American M1903. This legend is also false, but it endures, and every time there's a new generation of military rifles some dolt starts repeating it again for no discernible reason. I imagine you heard your story from one such dolt.
The American AR-15 and its variants (M-16, M-4, etc.) are chambered in 5.56x45, AKA .223 Remington. The Russian AK-47 and its equivalents (including the AKM, etc.) are chambered in 7.62x39, which is a completely different cartridge with a completely different size. The 5.56x45 is a much longer cartridge, for a start, and also thinner. If you dropped one in the chamber of an AK like a dummy the bolt wouldn't even close all the way. It would be impossible to fire the gun out of battery like that, but if you somehow managed it the casing of the 5.56 would surely explode because it is not contained by the walls of the chamber. You couldn't even begin to fit a 7.62 in an AR-15's chamber. The cartridge is too fat. I think you'd have trouble fitting it through the ejection port, and you can forget about jamming one in the magazine. It just ain't gonna happen.
Likewise, the modern Russian AK-74 and its variants are chambered in 5.45x39, which is superficially similar in concept to the American 5.56x45, but is still a completely different size. Again it is a shorter cartridge and this time with a smaller diameter bullet. A 5.56x45 cartridge will be too long to chamber in an AK-74, and too fat for the bullet to fit down the barrel. A Russian 5.45 round dropped in an AR-15 would just rattle around in the chamber. Again, if you managed to set it off somehow it would just explode in place, because the casing doesn't fit the chamber properly.
I think some of the confusion comes from the fact that you can modify an AR-15 variant rifle -- by way of a major parts swap consisting of replacing the barrel, chamber, bolt, and attached upper receiver assembly) to fire different calibers, up to and including Russian 7.62x39. The vast majority of upscaled AR-15's are actually chambered to accept .308 Winchester (AKA 7.62x51) which is again a totally different cartridge than 7.62x39 Russian. Don't get confused by the 7.62 in both of them: The .308 Winchester is longer and stouter than the AK cartridge, and plain old will not fit in an AK-47, and vise versa.
The only result of making your rifle's chamber half a mil bigger than the enemy's ammo so you can physically fit his cartridges but he can't fit yours has no effect other than allowing the enemy's ammo to explode in your gun's chamber. This has no practical advantage at the expense of making your gun woefully unsafe to fire with the wrong ammunition in it, which you purposely designed it to be able to accept.
TL;DR: Guns do not work that way. People are confused enough about firearms as it is, so don't contribute to the problem by perpetuating falsehoods like this. It only leads to some redneck dim bulb trying to shoot a 7.62 out of his .223 deer rifle, maiming himself, and therefore causing some politician to pass a law about it that makes life harder for the rest of us.