N.H. Rev. Stat. Ann. § 106-C:4

Whenever pursuant to the authority granted under this chapter, the members of a local police department, or of the state police or the sheriff's department are engaged in assisting another county or municipality in combatting an emergency in such other county or municipality, such police department personnel shall possess the same powers, and have the same duties, rights, privileges and immunities they would have if they were performing the said duties for the county or counties, municipality or municipalities to whose sheriff's or police department they are normally attached.