Ok here is my experience..

I did my undergrad from US and applied directly to the Phd program. I applied to several schools and in all of those I checked the box that asked for funding so basically I said I will need funding from the school. When I was admitted to the university that I am currently enrolled I got my Graduate Assistantship letter together with my acceptance letter into the program. In my I-20 it said that my tuition will be paid by school and I will be working as a Graduate Assistant for which I will get a stipend and I should use that stipend for my personal expenses (Room and Board). As far as I make satisfactory progress I will be getting funding from the department for 5 years. However, some of my friends from other countries were admitted without funding. They used their personal funds for the first semester  and after that they applied for funding and some of them got it from my department itself and some applied to other departments and got funding. However, for them funding is not guaranteed every semester. They just have to apply and see how it goes. Also in my program we have to take  qualifying exam after we finish our first year. Only if we pass that exam we will be allowed to continue in the program and will get funding from the department. If we fail that exam then we have to withdraw from the program with or without a  masters degree.

In our program if we get funding from the department then ..first year we have to work as a GA which is basically grading hw and exams for a professor...from second year we either work as a TA or AI...TA means basically running review/discussion sessions for a group of 30-40 students..grading hw/exams...AI (Assoicate Instructor) means teaching your own class..so you have to do all the work..give lecture, prepare exams, grade hw and exams..hold office hrs and everything else..Salary for AI is higher than GA and TA..In order to qualify for AI we have to take this English speaking test..which is pretty stupid.. :-)

I hope this answers your question.

Goodluck