For me the term local contrast involves the localized contrast within a given tonal scale on the print. Local contrast is what others and myself consider necessary to give a print the glow possible with black and white materials. So to answer your question yes one would want high local contrast.
We can achieve local contrast by increasing the paper grade or increasing the filtration in the case of VC materials. However when we do this the overall contrast can and very often does exceed the scale of the paper. This is why printing methods such as flashing the paper and contrast masking are utilized. These methods allow one to contract the overall contrast while expanding the local contrast.
Printing of fine images requires a balancing act of achieving local contrast while not exceeding the overall contrast of the paper on which one is printing.