It is of general knowledge that one can (and even need) to apply for scholarship to attend graduate school. AMINEF is well-known as the main source for scholarship to study in US.
It contains significant amount of awards specifically for Indonesian pursuing graduate degrees. This however comes with high competition since the number of applicants are substantially higher than the available awards. Sadly many Indonesians rely on this as their source of scholarship and hence, possibility of pursuing a graduate degree.
It depends on the field of study. Fullbright-Sampoerna Foundation MBA scholarship is one of the most to be envied. It provides the rarely available scholarship for those seeking an MBA degree. For other subjects however, it is not always necessary to apply for such scholarship. Strangely it remains unknown to many that, for example, getting a PhD in US requires almost no external scholarship. Due to the nature of the fields, it is not uncommon for students admitted to prestigious schools to be offered financial aids, in the form of research assistantship (RA) or teaching assistantship (TA). Examples of such financial aid can be found here: example 1 and example 2. It is as simple as ticking ‘yes, I need financial aid’ in your application forms. Naturally, you need to be competitive enought to be admitted.
Regardless of possibility to received financial aid from the school, being a Fullbright scholar carry its own prestige. So go ahead, test yourself and see if you worth the title. Just don’t put your eggs in one basket.