There are some quite harsh comments here, leaving them alone is about the worst thing for everyone.
I was a shy child and although encouraged by my mother, missed out on a lot of fun.
My 8 yr old is no longer shy in most situations.
Firstly I banned the word shy and talked about what was he actually worried about e.g newness, bigger kids.
I acted confident with strangers and said hello/started short conversations with random people/ shop workers at all times (lead by example)
I talked up the benefits of clubs for months in preparation and pointed out that being new places and situations are a bit scary for everyone, even if they don't look like it bothers them.
Pointing out that at the end of an hour these surroundings will look so familiar... By next session we'll know the place so well it will feel like home etc
At parties i'd tell him if he didn't play and just clung we'd just leave after a set time (say 25 mins) be matter of fact and stick to your word. Give top up cuddles and let them know you'll be there throughout or tell them when you intend to leave.
When he was ready for a club we sat in on two sessions and on the last the leader held him back while we left. When the door shut behind us he went off and got involved.
We have such a confident boy now I marvel (secretly i'm still very shy) we never draw attention to how he used to be as kids love their story.
If their story is that they're the shy one, they'll live up to it! They need to be gently encouraged and basically coached into confidence. It can be acquired.