A place-based social network is similar to Facebook or Twitter in functionality, but is focused on curating the civic experience of a relatively small locality such as a city or neighborhood. It differs from place-based groups on larger social networks in that the entire experience is organized around a particular locality.
There is such a social network in the form of NextDoor; however, as critiqued by Pete Davis, this social network is primarily focused on complaints and crime remediation, rather than any kind of civic organizing or larger civic life. Davis describes the process of building a place-based social network in some detail in his Great Battlefield episode at the 9:17 mark. He describes the idea as something akin to a “Wordpress install” for a town that could be installed to a local server, maintained by a civic organization like a library. The key to his idea is that local communities should be akin to a Redit forum, where everyone has the same feed, and there should be maximal public space.
Ethan Zuckerman describes a related concept for an online town square in his Great Battlefield episode, at the 41:31 mark. This project is described in some detail at the Initiative for Public Infrastructure as well.