I owned a dam. Worst deal of my life, as it was regulated by the town, state, and feds.

Touching anything on it required permits from multiple agencies at each of the above.

Be careful doing ANYTHING with water, as that is regulated by feds, states, and ofen localities. Wetland regulations are extremely restrictive. And wetlands are often not wet! (Depending on where you live, state and local often try to "out green" each other by layering on even more restrictive regulations.)

"Is it actually legal to build a private dam? "

Not likely. Unless it is a very small stream.