Mostly the waterways are crown land out to the neighbouring private land.
Private owners control access to their land only.
If there's a gap between the neighbouring deed boundary and the wateredge, the private owners of the neighbouring land have access to the land in that gap to water edge for themselves/cattle etc.
The NT Water Act lays out NT Govt's laws.
I'd say that the property owner can authorise you to access his land but not to access the crown land between his boundary and the water's edge. He, his family and workers can access it under the Water Act but he can't can't authorise others to enter crown land.
In the case of TO Land (classed as private land), the NT Aboriginal Lands Act ( its Federal legislation) over-rides the NT Water Act where they're in conflict.
So the TO's can authorise you to access their land even to the water's edge. Probably for a fee or via a land use agreement.
Under the Water Act you can probably travel up the river/creek by water without needing permission but if the river is part of a TO claim via the Fed Govt's NT lands Right Act, control of the river passes to the TOs. Then you need permission to access it (irrespective of what the NT Govt thinks) and fees probably raise their head again.
Just like the owner of land under NT Water Act, the TO owner of that land under the NTLRAct isn't obliged to allow you access.
That's my view anyway.