The issue of grandparents living independently is partly a question of traditions in many different cultures and nationalities. However, there also seem to be problems of affordability, and thus it is rather complex question.
On the one hand, those who support the idea of elderly people living with their children and grandchildren together point to the higher level of emotional support in such situations, for example in times of crisis such as bereavement. Furthermore, grandparents may bring up their offspring, in cases when parents do not have enough free time, for instance by telling stories such as Cinderella and Robin Hood they instill the sense of kindness into grandchildren. Finally, living with extended family provides a good security, as older members can be guards against accidents and even robbery.
By contrast, supporters of grandparents living separately rely on the increase of self-reliance, as this will boost self-esteem and will prevent mental decline in old age. It is also true that many younger families may not have enough housing space, or indeed spare time, which is necessary to take proper care of elderly people. As a last point, some grandparents may wish to live separately and having their relatives visiting them, as it makes meetings more special and cherished for all concerned.
Overall, the stronger argument to me is in favour of grandparents living independently, with all the advantages of self-reliance and separate space. This is provided that their health and finances allow them to continue living separately, without the risk or fear of isolation.