The answer is, Yes, No, It Depends, in true agile fashion. :)
So I've been lucky or unlucky as the case may be and have never worked in a company with the Business Analyst role. So I can't comment on Zsolt's answer.
What I do know is that in the agile community we are seeing more and more the concept of the Product Owner Team. And the concept of what I call Pyramid Planning is still very valid in agile.
Pyramid Planning: The idea of several stages of planning with greater levels of detail. The large scale agile model, SAFe, uses this extensively. At the highest level prioritization is done by a business/ strategic team. At my present company (AOL) we us this regularly. As you get closer to the actual development work, the people involved in specific planning change. VPs and Execs decide the four big things that will be done this year, the team will decide just what gets done for the iteration next month. Which brings us to the idea of...
Product Owner Team: Having been a product manager, in a past life, I can say without a doubt that few, if any, product managers get to make all the decisions themselves. They are usually little more that the cowboy desperately trying to hold onto the bucking horse and guide it in the right direction. The Product Owner Team (POT) recognizes this and formalizes it. The POT is made up of the key individuals who can provide the PO with all the data to make an informed decision. A POT will often have, in addition to the PO, an architect, representative of the development team, Ops/IT, customer support and BAs if the company has them. As a team they will review User Stories and prioritize them based on the entire picture, not just a limited business view that often happens if a Product Manager tries to work in a vacuum.
And it is important to note that you need to always engage with the team to get at least a high level estimate for the level of effort. You, the product owner, may think teleporting cars is the most important feature ever. And you may be right, only the dev team will tell you that no matter how valuable it is, it will take a 100 times longer than making a new self-driving car. You always need to factor time into your prioritization and only the team can give you a realistic estimate.