Nope, it's not a requirement... There's always the chance that you get "discovered" while doing something completely different, or you could show up to casting auditions without having any formal training.
However, studying drama or acting can be very helpful because:
You get some practice and insight into what it takes.
You'll learn skills and techniques from people who know about these things.
You'll learn the terminology and basics which people who show up to auditions are expected to know.
You'll learn practical things like how to create a show reel, what kind of photos and information you should send for potential roles, how to market your talents...
Maybe most importantly, it'll be an opportunity to make useful connections in the industry.