Who is considered Sayy'id and can this be proved?

As I know Say'id is descendant of the Prophet Muhammad (SAW) but can anyone prove that they are one? The names Mir and Saeed/Sayeed/Said are usually used by those who claim to be the descendants of the Prophet. Can anyone have these names or not? Say'id are now in nearly every single country. Is this possible? Has this been mentioned in the history of Islam?