I met an independent researcher (Bob Edgar) who had sold his company to Intel and was bored. He started to attend seminars at Berkeley to find people who wanted his help, met a few folks, and ended up being one of the top people in the field of protein sequence alignment. At first, my advisor helped him by giving him an arbitrary appointment at Berkeley because otherwise journals wouldnt accept his papers because he was "independent". Eventually he reached the point where he could publish with his email address as affiliation (https://www.ncbi.nlm.nih.gov/pubmed/15034147). Of course, these days, he'd just publish to biorxiv and put the code in github.
Reading his CV (http://www.drive5.com/papers/resume.html), it seems that even though he is a peer reviewed "Computational Biologist", his PhD is in Theoretical High-Energy Physics, who became a systems developer.
I’ve always thought that an independent researcher was someone who studied within their field just had no monetary backing. Ex: a person with a PhD in X that does research on a subset of X, in their free time.