Water softeners don't use salt to soften the water, they use salt to clean the hardness off the negatively charged plastic beads that the water flows through. At night, when the water isn't being used, the water supply is temporarily used to flush water through the salt block and basically rinse out the beads, which is then drained.
You should only get a trace amount of sodium (milligrams per cup of water I imagine) just from any tiny bit of salty water that wasn't backwashed out during the rinse cycle, but nothing more. If your water has salty taste that means you probably need a new water softener
I thought water softners typically used an "ion exchange resin"[1] where the hard water (magnesium, calcium) ions are swapped for sodium ions[2]. So you should be able to taste some sodium.
I was under the impression its simply best practice to not drink water from the shower/hot faucet. I've never tasted salt myself, just something I'd been told over the years.
The main reason not to drink hot water is that it can be sitting around in a tank for a while, and it isn't held at a temperature which kills all bacteria.
