I do not recommend doing this.
Each time I washed it salt would recrystallize as it dried, so after 3 washes I decided to soak it in a lot of water which resolved that.
Then I went to use the salt and without any salt build up it poured out way quicker and ruined my breakfast.
The solution (heh) here is to pour some salt into a dish or your hand for salting the water, instead of exposing the shaker to steam.