From what I understand Team USA athletes are paid a salary/stipend by their respective federations (i.e. USA hockey) for training and receive bonuses for medaling. However, many of the athletes do seek sponsorships or work full/part-time jobs on top of their training. So in this context they could be both amateur, in the sense that they still may have to work to support themselves, and professional, being paid to train, at the same time. Of course some of the more famous athletes will go on to sign endorsement deals and so on (Ohno, Phelps, Vonn, etc.) and I would imagine a number of the athletes compete in different non-Olympic tournaments for cash prizes.
I'm not sure how it works for every other nation, nor am I sure the level of compensation each USA federation offers their athletes. Plus, you could have athletes that are already wealthy that seek training regiments on their own dime.