We don't yet know what precisely the "local" benefit is. Many of the relevant studies investigate the effects of local newspapers closing (or opening, rarely). But they imply that there is something special about local reporting that can't be supplied by national news.
The best answer comes from a ~10 year old study [1] that suggests there is a mutual reinforcing effect of local news, community embedding (i.e. social capital) and social engagement (i.e. volunteering etc.)
I wish there was a better model of (1) what it is in particular in local news that is beneficial and (2) how it affects people (positively and negatively).
[1] Rojas, H., Shah, D. V., & Friedland, L. A. (2011). A communicative approach to social capital. Journal of Communication, 61(4), 689–712.