The primary responsibility of a product manager is to be the voice of the market. This requires experience in a domain, curiosity, empathy for people and their problems, skill to conduct market research, the ability articulate a product to the people who build it and the ability to articulate the meaning of the product to the people who buy it.
Engineers who have worked in a domain for a long time become product managers. They take the experience route. User Interaction designers or product designers who work with other product managers gain domain experience and go on to become product managers. Product designers, typically with interaction design or industrial design background can bring important skills such as user centric design and similar research methodologies to the table. Engineers and MBAs are not trained in such methodologies in college.
Companies hire MBAs right out of college to manage a product because of their quantitative market research skills. Such MBA's normally, though not always, tend to take the product to the market first and then slowly and steadily gain domain knowledge, gain ability to conduct customer research and, sometimes, the ability to appreciate good design.
When I first interviewed for a product manager job, I got the interview because I had an MBA. MBA will open doors for you. But an MBA may not get you the job. I was hired because I had more than 15 years of domain knowledge in the field of human capital management.
The two day course "Practical Product Management" conducted by Pragmatic Marketing is a good course to attend to bolster your chances before applying for product management jobs. The course will give both engineers and product designers the edge they need.
The ideal way to step into product management is to find a product manager who is willing to mentor you, preferably on the job.
There is a school of thought now-a-days that says that the ability to design and the ability to make things with your hand are far more important than the ability to analyze, prioritize and manage. I believe thinkers need makers and vice versa. If you can think and make then you are in the Mark Zuckerberg, Bill Gates category.