If you are in the US, you know that you cannot volunteer for jury-duty even though most people hate the idea of having to do jury-duty. It is similar with reviewing. We assume it is better if people get drafted into reviewing.
How do Associate Editors (AE) select reviewers? Usually by looking a people who wrote papers in the field that are knowledgable about the paper. Some journals have editor assistance software that tries to find these people for the associate editors such as MDPI. Usually, an AE reads through the submitted paper and places special attention on the papers cited, as the authors of these papers should be knowledgable, especially if the papers have a good citation count themselves or a published in difficult to get in outlets. If it is a conference, they usually first form a program committee and then papers get reviewed by the committee. Members of the program committee can ask others to review. It's supposed to be highly co-operative. Journals keep tracks of reviewers and ask the AE to grade the reviews. Someone giving poor quality reviews is not likely to be invited again.
If you want to review, you are best of publishing papers in reasonable outlets in your field. It used to be that professors would give their Ph.D. students papers to review (for which they themselves were invited to review) and then discuss the paper with them and the student's review. This is now considered to be a breach of reviewer confidentially if it is done without the knowledge of the AE, but at least it taught students to write good reviews and in the process learn what a good paper is like. This also helped the students to write better papers themselves, of course under guidance and supervision. Being an independent researcher means of course that you are cut of from this type of training.
To get invited to review you first need to have some (good) publications under your belt. If people active in conference organization and journal editing know about you, they might ask you to get involved. It is the quality of your publications that ultimately qualify you as a reviewer. Once you are considered "reviewer material", if you accept the invitation to review, make sure that you submit a good review in a timely manner and communicate with the AE if there are any problems.