No they don't, not in my home country. Fulbright terms and conditions change according to the country, each country has a different arrangement. Mine required me to apply to a terminal master's, and I did so. There were others who applied to the PhD scholarship initoally, which is also given for two years, but I chose the masters scholarship. So, these two are different.
They don't have any preferences for the type of school (public or private), but they only give 50k at maximum, thankfully I received tuition remissions so I do not have any problem in this case.
Thanks for your answer but I know the terms and conditions of my scholarship. In the end, it was I who read all the documents, discussed the terms with the commission, and put my signature eventually. Here, I am asking about the schools, and not my situation