Most volunteers take some US dollars with them. It's a good idea, but not absolutely necessary. In 1996, right before my first tour, I was a very poor recent college graduate. I had emptied out all of my savings and summer earnings paying off college debt. I spent my last cash on an expensive sleeping bag and hiking boots for Nepal. When I got on the plane for staging, I had about $100 in my pocket. I didn't even have a bank account anywhere. And I was fine. Peace Corps gave me enough money at staging and during training and service. On the other hand, I didn't have the money to travel much during service. A lot of volunteers went to Thailand and Tibet for vacation, which I couldn't afford. I had to go to India because it was cheap. India is exotic, but disturbing in some ways and thus not very relaxing.
In my opinion, the Peace Corps living allowance is adequate. I had enough money at the end of service to buy a couple of gifts to take home and even to buy my plane tickets home in local currency. But I have to say that I am a fairly frugal person with a low maintenance lifestyle. As a volunteer, I rarely went into the capital city, which is where most volunteers burn through their cash. I knew other volunteers who were regularly low on funds and cashed their own dollars to get by or had family members send them money from the U.S.
During my service in the Dominican Republic, the peso devalued dramatically and there was rampant inflation. Prices went up every week. Unfortunately, Peace Corps is slow about raising living allowances. Peace Corps is a big bureaucratic government agency. They have lots of rules. One rule is that living allowances can't be raised unless volunteers turn in a survey about their monthly expenditures. A certain percentage of volunteers must turn in the survey for the living allowance to go up, and the percentage is never reached. One quarter of volunteers are always about to finish their service and know the change will be too late to help them, some volunteers never get the survey, and still others are too lazy to send it into the office. The country director complains about not receiving the cost of living surveys from enough volunteers, but eventually gives in and raises the living allowance anyway. Everyone is cozy until the local currency devalues another 50%.
In any case, your living allowance is not tied to the U.S. dollar so in countries with bad economies, you feel all the same bumps and bruises that the nationals feel. Watching your spending power deteriorate day by day is quite educational. There is an American attitude that people are responsible for their own financial destinies. It turns out not to be true. You can stash every peso you earn in the bank for ten years and be on the verge buying a farm to make yourself agriculturally self-sufficient; but if your government is run by crooks (many governments are) they might be funneling large portions of their country's foreign currency reserves into their personal Swiss bank accounts, as a result of which all that money you saved for a decade is not going to buy you a little potato patch by the river, never mind the farm you wanted. I am not an economist and don't understand such things on more than a superficial level. But I have lived on a modest salary paid in a sub-par currency, and it certainly gave me a new appreciation for the mighty US dollar.