Bit of advice for anyone trying to get an REU and hitting the wall. First two years, counselor suggested my kiddo apply to five she liked…which she did… and didn’t get any of them. This past year, she applied to 22, and got two offers. Sometimes it really makes a difference to put in the extra effort and expand your application pool. Also, don’t give up. Third year was the trick. Freshman and sophomore years she spent working as a lab assistant at a biological station, which wasn’t as cool as participating in her own research, but helped her learn a lot of useful techniques and lab skills. Pouring plates, running PCRs and doing lab dishes isn’t glamorous work, but it builds bridges and shows you have a good work ethic.
My impression is that REUs aren’t always your own research – you are still working to advance the research of the prof that runs the lab. Not that they aren’t valuable, because they are! But they aren’t really driven by the student in most cases, I think.
It depends very much on the lab. There will certainly be suggested areas of study (sometimes very specific areas), or continuations of previous projects. A lab might also consider your ideas for an original project, particularly if it ties in. But yes, some REUs do produce publishable research. If you’re listed as an author, by definition it’s your research. But you are correct that REUs do not always offer original projects and that research is almost always collaborative.