The buses are shared by the hotels in the DD area, about 6 or 7 of them. The bus stops at each hotel going to and coming from the parks. They are not even close to being as regular as the Disney Transportation system. They will keep stopping and picking up people until the aisle is standing FULL.
Don't get me wrong, I have stood on the Disney buses too, but not as often and usually only on returning to the resort after park closing.
Also, the post about being picked up by the DD bus in the 'middle' of the parking lot isn't far off! And that's not fun when it's dark and you are standing somewhere out in the lot hoping you are standing in the right space for your bus.
If you are buying park passes, you can walk over to DD to the bus area and get the Disney buses to the parks, etc. They are all marked. When we stayed at the Grosvenor, that's what I did after the first day. Of course when you are going to MK from DD, I believe you go to the TTC instead of the MK entrance.
Maybe someone else can tell you for sure because when we were at the Gr. it was in 1999.