my n=1: I am on an extremely low sodium diet (around 500mg/day) and I didn't find any difference in thirst or hunger levels compared to when I was on a more typical 4000-5000mg/day one. The only change for me (besides of course a period of adjustment where food didn't taste like much) has been a significantly lower blood pressure (from typically 145/85 to 105/65), everything else has remained pretty much the same.
My understanding has been that some people are more sensitive to sodium than others. If your BP is normal, chances are it won't have much affect, but if you've high BP then it is more likely to have an affect on your BP. I can't comment on causes, perhaps those who are more sensitive are more likely to have raised BP.