From what I can gather, this has the same scentence structure French does, without all of the words. (The house of my friend vs. My friend's house) They just say "She student my friend". I guess of you look at it and see that this is what the sentence portrays you can work it out?
Possession in Russian is expressed through the genitive case -- моего друга is genitive, so this sentence is absolutely unambiguous. You can look at it in opposition to, for example, "He's my [female] student's friend" --> Он друг моей ученицы. You're not entirely wrong in your comparison to French, as the genitive case here fills the same role as the preposition "de" does in "C'est l'ami de mon élève." -- you simply were missing a crucial bit of information :-)
Of course kpagcha's proposed translation is also correct, and it's almost certainly been added to the possible answers by the mods since then.
Свой and its declensions always refer to the subject of the sentence as the possesser (the subject is the noun or the pronoun in the nominative case), so here «Она ученица своего друга.» would mean "She is her [own] friend's teacher."
It's also true when you or I are the subject of the sentence, and this is where it gets hairy for us English speakers. For example, "I am my brother's keeper." in Russian becomes «Я хранитель своего брата.» and not «Я хранитель моего брата.» Likewise, "Mind your own business." becomes «Занимайся своим делом.» and not «Занимайся твоим делом.»