Is there a chance I can buy a cheap phone (and put some money on it) in Peru, so it will work in other countries such as Bolivia and Argentina? I'm not talking about technical issues, but about carrier issues? What phone carriers are in these countries and is there a "shared" carrier in these countries?
No each of these countries has a different set of carriers. Even where the name is similar such as Claro Peru and Claro Argentina the SIM chips do not work on each others networks.
However the solution is simple. Just buy a cheap unlocked phone here (most phones in South America that I have seen have been unlocked) and buy a new pre-paid SIM chip in each country. They usually cost a few dollars for the chip plus whatever phone credit you want for the chip. You might need to have your passport or national ID card with you when you buy a chip or have a local friend buy it for you. This wasn't required in Peru or Bolivia but was when I was in Brazil and might be in Argentina.
Personally I got a quad band phone that it could work in any country in the world but if you want to save on the phone you could get a dual band (GSM Frequencies: 850/1900) one that would in in South America and maybe a few other places. Nextel phones only work on their network.
You can find info on cell phone companies and bands in other North and South American countries here http://en.wikipedia.org/wiki/List_of_mobile_network_operators_of_the_Americas