No, I do not think that state's attorney's should supervise local ones. This would lead to jurisdictional issues and possible conflicts of interest. It would definitely lead to clashes, which would tax an already overburdened local legal system. If you really want to ensure that local courts get the job done, they should be given the resources to do so rather than constantly facing cutbacks.
I think that there needs to be some sort of limited supervision of the locally elected District Attorney, whether it be by the State Attorney General or someone else. To me it makes sense that the State AG's office have some authority over the local Attorney
In my opinion, there is no reason that this should be done. As long as the locally elected district attorneys do their jobs in a way that is A) legal and B) satisfying to their constituents, there is no need to have the state attorneys supervise them.
Basically, the reason for this is the same as the reason for federalism. It makes more sense to have people be governed by the government that is closest to them. This is because the closer levels of government are more aware of the feelings and needs of the people.
Following this idea, locally elected DAs would have a better idea as to the priorities of their communities. They would know what their constituents think is important much better than a far-off state attorney would.