In the case of a natural surface (wood chips or bare earth), e-bikes are not permitted on trails designated for non-motorized vehicles unless the owner of the facility, such as the local or county government, specifically allows e-bikes.
The rules for paved bikeways and shared-use paths are the opposite; e-bikes and e-scooters are, by default, permitted on paved paths where traditional bikes are allowed, but local or county governments have the authority to restrict them.
On state-managed facilities, the New Jersey Department of Environmental Protection allows class 1 and 2 e-bikes on improved multi-use trails with hard surfaces such as crushed stone or asphalt that are suitable for both e-bikes and traditional bicycles.
E-bikes are permitted in all national parks on trails open to pedal bicycles, with minor exceptions.
Since the New Jersey law specifies that local governments can restrict the use of class 1 and class 2 e-bikes on trails, even if traditional pedal bikes are allowed, be sure to check local rules and regulations before taking your e-bike or e-scooter on a trail or a boardwalk.