A lot of parents will cut a shy child plenty of slack and put off trying to help them, when they are sensitive to the child's discomfort. I honestly think this is misguided, and teaching plus rehearsing will go a long way towards helping a shy and self conscious child enjoy life more.
It's important to acknowledge the starting point and to emphasise that there is nothing wrong with being shy and that everybody feels like this to some extent, some more, some less. A person who is very self conscious can (ironically) sometimes lose track of the fact that there are other people in the equation however, so teaching involves talking about why other people are talking to the child, and about their expectations in terms of response, while at the same time acknowledging that response is difficult.
My DD1 couldn't see the point of chit chat but didn't realise it is how more interesting conversations get off the ground. She was impatient with it, thought people were just being stupid, and didn't realise that you don't launch straight into a conversation about things you are interested in without exchanging verbal gestures of friendliness. She also didn't understand the importance of sending signals of friendliness or willingness to chat. This is not a child who was autistic; she just had a combination of shyness and seriousness.
In a way, I always felt this awkwardness is more noteworthy in a girl than it might have been in DS -- many people expect girls to be more social than many of them are. But all the same, I did not think I was doing DD1 any favours by having her unwittingly carry on a crusade against preconceived gender expectations, though I guarded against turning her into Little Miss Conviviality. We talked about her reluctance to speak or acknowledge greetings and her feelings about this, as well as whether she wanted to learn to do things differently.
We rehearsed a few stock phrases in a playful manner as well as tones of voice that she could use convincingly in response to 'How is school going?' or 'How's life, Tootsie?' or 'Would you like another slice, dear?' -- the sort of questions that normally left her resembling the verbal equivalent of a beached whale. She didn't understand that a three page essay was not needed in response to the first two questions or that a simple and audible 'yes please' or 'no thank you' with appropriate facial expression would do fine for the third. Optional additions such as 'It was delicious but I am full' were also rehearsed.
We also did a little role play where DD1 was the person initiating a conversation and I was DD1 giving her the brushoff. I asked her how she felt when I seemed to ignore her. Did she want to keep on talking to me? Did she feel like calling out 'Hi!' when she saw me again if I had stared hard at the ground when she did it the first time?
Because she had told me that she preferred to feel that she was invisible and unnoticed and to just listen to other people, and seemed to dislike the idea of a spotlight on her, I encouraged her to see interactions as practices, not earth shattering debuts from her preferred state of being a chrysalis into the persona of a social butterfly. So if we were going to visit someone or if a visitor was expected, I would say 'Great, you can practice on them'. She kept on 'practicing' until having a pat response and audible conversational voice became second nature to her.