In a word: networking.
Most conferences and journals don't have any formal process for people to offer themselves in this manner. I'm not exactly sure why not, because most publication venues are also always wishing they had more trustworthy reviewing capacity. I think it's the "trustworthy" part that's the issue: there is a notion that the people in charge of selecting reviewers should have some reason to believe the review you turn in will be a meaningful evaluation of the work under consideration. Still, in practice I often identify possible reviewers by looking at their online presence, so having a person submit themselves as a possible reviewer seems like it would be no less reliable--- — it's just that in my experience it's just not done.
Instead, the most frequent wayways that I see people getting asked to be reviewers is 1) by having people get to know them and think of them when they're looking for people to review, and 2) by having other people who are declining to review point the editor to them as an alternative. If
- by having people get to know them and think of them when they're looking for people to review, and
- by having other people who are declining to review point the editor to them as an alternative.
If you put the word out into your professional network that you are looking to review papers, then reviews will likely start coming your way. Furthermore, once you've reviewed for somebody, you'll be in their database: provide a high-quality and timely review and fill out the reviewer account with keywords and such, and you'll be more likely to be tapped again to review for the same venues, because reliable reviewers are like gold.
