Here's a guess: No. They build them, they maintain them. I am sure that they pay property taxes on the lands on each riverbank. In the case of the Woodsville - Wells River bridge, the situation is clouded by the fact that it was a combined rail and highway bridge, so I'm not sure of its status.
A little more background: some time in the middle 'nineties, the bridge was determined unsafe for rail operations. At the time, the B&M had already given up the Woodsville - Littleton segment (by lease, IIRC) and that was being served by the NHVT. The condemnation action effectively cut off NHVT access to the south via White River Junction, so in compensation Guilford gave them rail access via St. Johnsbury to reach the Connecticut River route. With no further need for the through route and little or no on-line business on the New Hampshire side, that route was abandoned and torn up in the late 'nineties.
PBM