Organizations that don't charge fees for volunteering, such as VSO Canada, UN Volunteers and the PeaceCorps are looking for people who can work (read, write and speak) in a language other than English, who have skills and experience that can lead to local people generating income, better feeding their families, improving children's health, etc., or that can lead to the transformation of key institutions, such as government, universities/schools or NGOs. They are looking for people who can commit to a two-year assignment, who have experience working with under-served communities, or who have a lot of experience in very diverse or religiously-conservative communities. The average age of volunteers in these organizations is over 30 (for UNVs, it's 38) and most not only have degrees, they also have grad degrees.
The goal is to give local people jobs and to keep money local, not to give Westerners a feel-good vacation -- hence why short-term placement agencies charge volunteers, or require these volunteers to pay their own way (flights, in-country transportation, health insurance, accommodation, food, security, translators, training, staff to supervise and support them in their service, liaisons with the police and local officials, etc.).
Here is a web site that can help you learn more about the skills and experience desired by long-term placement organizations, and how you can gain that experience locally.
There is a listing of the more-than-30 member organizations of the International Volunteers Program Association (IVPA) that is a good place to find reputable volunteer-for-a-fee programs -- programs where you don't need to have much experience in order to participate, and the placements are just for a few weeks or months:
The cheapest overseas volunteer-sending organization in the USA that will send unskilled volunteers for short-term assignments (six months or less) that I've found is SCI - Service Civil International ((http://www.sci-ivs.org/).
I know *nothing* about them other than what's on their web site, so please don't consider the listing of this organization in my answer as an endorsement.