It's always the "it depends" kind of answer. In bigger companies, you have teams dedicated to API governance, whatever they say, it's how it's done. In early-stage startups, you have full-stack folks doing end-to-end design, test, integrate with UI, document, etc. The trickiest ones are the ones that are in between these two stages. It's where it gets messy, and you MUST have something in place to serve as an API contract. Swagger dominates that, imho, Postman collections and docs are used a bit differently in my experience.

I'm helping build https://voiden.md which serves as a unified place for API spec, test, and documentation, but there are also plenty of API tools that focus solely on endpoint execution.