I don't know anything specific about shifting from an engineering role to product management one; but in my career I shifted from a sales job to marketing management then shifted to graphic design and lastly to writing, and found that the action I took to make the shifts are similar(except the graphic design — my second college degree), so I would like to share my two cents.

Every role requires different skills but if you have been in the job market for a while, you probably also have some of those skills already. It is important to research the role you'd like to have and objectively analyze the skills you lack and the proficiencies you have that you can lean on.

For example I know sales, marketing, and graphic design, and that makes me one hell of a copywriter. There are better writers out there for sure, but I can edit my writing a million times to make it look decent. Yet, my understanding of sales and marketing trumps the thought processes of many copywriters, and my design background makes me understand how the text I have written can be utilized. I can also help with either one of these skills in an emergency.

I don't wait for somebody to give me the job. As some likeminded individuals advised in the comments, as soon as I am sure that I want to change what I am doing as a job, I start doing the tasks related to that job. When I was a graphic designer, I started offering changes to the copy we are getting. I started working closely with the editors and made sure that they know that they can trust me to help them. I became their go to guy in the department when they need help.

I work hard to get the necessary skills for the task. I spend my private time to master the tasks that the new occupation requires.

I market myself, my desires, my loyalty and my skills. Show that you want to move towards that area and also that you are working hard to learn about it. If you want to stay at the company you are working at, make sure that you say that you love the company, but you think that the other role might be more suitable for you.

I change the work of the other department for the better in ways that only I can. Provide a lot of value in the other occupation you'd like to do for free. Do the basic tasks nobody wants to do, help them solve their bottlenecks. While doing that do your normal job perfectly as well. Do not get demotivated. Just do both jobs at the same time and show everyone that you are capable.

If you do this for some time, the management starts to see that you are working hard on improving the company and an asset. In my experience, this alone is not enough to make the change, because usually there is someone else doing the job already, and you might have to work doing part of their job for a long time before they resign or get fired and management(whom you have been showing your desire and skill) might consider you for that position. That's rare and I wouldn't depend on it.

But don't be hopeless. You are coming from another occupation and you probably have skills others lack. Find out what is your unique proposition. Mine was English. The company I work for has to communicate with a lot of expats and ambassadors and even though we have a huge translation department, nobody could write eloquently enough in English. I could communicate in English to a certain degree and I can also do all the other tasks that are required from copywriters. Voila! Matchmaking in heaven!