G. Mark’s Answer
It's difficult to generalize from my own career. I started in a company that operated under the philosophy that engineers were interchangeable. Specifically, they wanted engineers and scientists in this particular arm to be flexible enough to do any part of a project from problem definition to requirements to design and implementation and management. I was lucky enough to have opportunities to work in all those areas. Particularly for problems that I'd gotten patents for solving, I also traveled the country and did presentations while also having the opportunity to interface with customers to get more insight and problems to solve.
In general, most engineers and scientists in my experience have tended to focus on one or a few of those areas at any point in their career. When I was investigating problems, I was also very fortunate to have access to labs and tools to begin prototyping and at one point, building hardware to support the laboratory team I was managing, who happened to be extremely talented and driven. Many engineers will be working on teams. The current trend of "design thinking" and the like is again encouraging engineers and scientists to be multidisciplinary and work on teams with this same quality. This gives the team variety of points of view for solving problems. So as an engineer, you may end up working on any aspect of a project, either longterm, or for the duration of that particular project.
In computer engineering, I was primarily designing programs and implementing them. As I progressed in my career, I was afforded more latitude to choose where I worked in the process. In my particular case, I enjoyed working solutions from end to end -- as I said, from the customer contact to the problem definition all the way to testing, deployment and support, both sales and lab developments.
The bottom line is that engineers do a lot of different things, but each engineer may settle in a particular area best suited for them.