N.H. Rev. Stat. Ann. § 232:19

When a highway is laid out across a stream or body of water constituting the boundary between 2 towns the commissioners shall determine the line across the highway to which each town shall construct it, and describe the same in some definite manner, so that it may be readily ascertained; and, upon the acceptance of their report laying out the highway, such line shall be the true boundary between the towns at that place, for all purposes relating to the highway.