Who or what determines the responsibilities of a shared road between two municipalities? 
Normally when a road is on a shared boundary there is an agreement detailing who does the maintenance and has ownership (including the subsequent requirements for maintenance and associated CHIPS funding). There should be a copy of the agreement file with the clerk’s offices in each respective municipality.
Sections 182 and 187 of NYS Highway Law detail the legal issues on the matter.
Resources:
NYS Highway Law 182
https://www.nysenate.gov/legislation/laws/HAY/182
NYS Highway Law 187
https://www.nysenate.gov/legislation/laws/HAY/187
