That's one of my favorite things about bettas, their discretion
I'm sure you've seen betta poop, just not him in the act of doing so. I have seen mine poop before, usually after/during a large meal or when they get excited. They do it just like any other fish, but it's more chunky rather than long and stringy (you know what I mean)
I have tons of betta, and I've noticed that they're just like a dog or a cat...they have an area that they do their dirty business in, and in my experience ...it's usually behind a plant or somewhere where no one (the other bettas) can see/watch them. I love that! It makes for easy tank clean up because I know right where to go to find the major messes
There's nothing grosser than looking in a tank and seeing fish swim around carting a foot and a half of poo with them