Salt Water through a well becomes drinkable.
Besides, if there was salt water, the dwarves wouldn't be drinking from the river (as it would be salt water as well)
NOT ALWAYS. There's a lot of myths surrounding how salt water works.
I've run extensive tests in salt water biomes (and others)
In one fort I have a salt well on the same level as a fresh well, both are sourced from the same (river/fresh) cistern with no barrier. And dwarves used the salt well for washing, but not drinking, so it was correctly working, just not drinkable. It's possible to have salt-water biome underground in a fresh water area (river). Any water touching the salt-water biome is marked salty. THe "exact" mechanics of this are a mystery to me. In some circumstances saltwater spreads, and in others I have salt/fresh sources basically touching each other without mixing. Possibly the direction of flowing water affects this. The fresh well was "upstream" and the salt well was "downstream", possibly indicating saltiness of the tile is primarily calculated when a dry tile has water flow into it. I got not "back-contanimation" of the clean part of the resrvoir system.
Also, I made wells from salty murky pools - STILL SALTY. And pouring (bucket brigade) water from these into clean holes in the grass (which I had previously fillled with fresh water), disabled fresh water on the hole-in-the-ground permanently, backing up the idea that the wells were salty.
Screw pumps work better, this did actually clean the water.
My best theory at the moment, is that the tile you're drawing the water from affects the saltiness, e.g. so my wells were directly above the surface of the murky pool (tile labelled with salt water). Maybe if I built it another level higher, it would draw as fresh.
Screw pumps take the water and teleport it up 1-z-level, but also across horizontally. This did actually allow me to draw fresh water from the salty murky pool, and fill the grass pits with drinkable water.
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My advice for salt-water embarks is to create a screw pump on the surface of the salt-water source, make a tiny walled area it pumps to, with a ramp up. Check it's a "water source"
Next, Channel tiny holes in the ground, and bucket-fill from the clean source. These were still clean in my salt-water embark (it was ocean-side). So the grass/soil itself is not marked as salt, at least the the start.
Finally, try digging these little holes underground, bucket filling with clean water, to find a good spot for a well. You don't have to keep the whole reservoir from getting any salt, just the point where the dwarves draw off the water.
There's the odd WTF? I had when experimenting with salt water, but generally I think it's a tag on tile themselves, and 100% definitely you can transport saltiness to a clean tile via buckets, I found no way to un-salt these tiles ever again, they were next to and previously identical to other fake ponds which stayed fresh.
A corollary of this is - never flood your fortress / land with salt water if you ever want to build any reservoirs there.