If someone has social anxiety then I would have thought going on a first date would be a pretty horrendous situation to put themselves in.
With shyness, once you start talking to people you generally come out of your shell, whereas if you're socially anxious then, chances are, you'll spend that entire time feeling... well... anxious.
With regards to the "will they always be silent", no one could really answer that but the person themselves.
Anxiety, social or not, affects everyone differently.
If this person does indeed suffer with social anxiety then I wouldn't push it, or keep on at them to "go on a date".
You'd only be making them feel ten times worse.