Actually is not an I20 what I received, it is a DS2019, which is the I20 equivalent for a J1 visa. As I understand, in order to qualify for a J1 visa an institution has to sponsor you (give you some sort of money). In my case the sponsor is the university, which gives me the full tuition (but you can ask for a J1 if you get over half the tuition).
If you qualify for a J1 visa, you have to communicate the university your decision to apply for that visa and they will give you the corresponding form. Of course if you prefer an F1, you tell them and they will send you an I20.
I don't know if all univ can issue a DS2019 for the students to apply for a J1 visa. But the ones I was interested in mentioned the two possibilities.