The in-country decisions vary by country. Larger countries participating in the Fulbright program - Germany, UK - have actual Fulbright Commissions that oversee the Fulbright program both US-Country and Country-US. However, smaller countries - like those in sub-Saharan African for example - leave the decisions to the US Embassy in their country. And for countries without an Embassy, I believe the decision is made by that country's office of cultural affairs or some equivalent. In all the countries, the supervising agency contacts your affiliation to ensure the feasibility of your project, logistics, etc. I believe I read somewhere that the host country ranks the apps also with the top being recommended to get the grant, and a few selected as alternates.