how does someone begin this kind of career
From existing jobs by specializing in game design. Or by having a degree in it.
You did mention you started studying it. Is that a translation issue or something? I was under the impression that you are a student. A formal student in an institution where they give courses and you will end up having a degree.
Did you per chance mean self learning, by "studying"?
If so, entering chances might be better than in other fields, but try imagining a company hiring a self taught engineer. That's not gonna happen. You need a degree or work experience. And having side gigs on a self publishing site like Itch is not the work experience I am talking about. Professionals are rare here. Oh, there are some one-person-developers and even some commercially successful games. But game studios are rather rare.
And even if you find a small game studio here, chances are, they are so small, that a dedicated game designer is not what they seek. You would need to have other skills. A dedicated role of game designer maybe starts at studios with 10+ people. In other words, you need people to coordinate, to have the role of a coordinator ;-)
Due to the nature of that role, I am not sure, if that is done by freelancers, the way art can be commissioned. It's basically the thing that is the core of the game. Can't really outsource that. But other people will know more about this than me.