Yes, this is something that your software development method determines, unless it does not talk bout it; there is no right answer, and the better answer depends on your project and people.
The task that you are describing is the functional decomposition, where you take each requirement in the business analysts' requirements specification and break it up into the functionality that implements that requirement. This is also known as functional design, so you are talking about a functional designer.
You may choose to document your functional design directly in the description of the New Feature JIRA issues, as we sometimes do, or you may have an intermediate functional requirements specification. In the latter case, splitting up the functionality into developer tasks may be more straightforward (in the sense of not being a design activity) and might be done by the project manager or technical lead who assigns work to developers, or alternatively by developers who assign work to themselves.
In every case, this task is about translating from the business analyst's perspective to the developer's perspective. How you need to do this, and in how many steps, depends on how well the writer can put it in the developer's terms, and how well the developer can understand requirements written in the analyst's terms. Fewer steps cost less time and introduce less noise, but require people with better understanding and communication skills.