OP, I am speaking from a position of having two children with very different personalities. My older one sounds very like your DD (energetic, sociable, wild, loves to run and explore, needs guidance to behave appropriately sometimes) and my younger one more like your friend's DD (quiet, a bit shy, takes some time to warm up). Strangely enough, the only time my little DD is truly wild is when she's with her big brother!
So I don't see this as a parenting issue like everyone else seems to. When you have a child with a certain personality and it's your first child, you tend to see that as being 'normal' and 'correct' and you don't necessarily understand the other child and where they're coming from in the way you would if you had more experience.
When I only had DC1, I had a friend who had a child much like your friend's child. I came to dread meet ups, as her child could do no wrong in her eyes and they would turn into a constant commentary on what my child was doing wrong... running, climbing, not playing the game the child wanted, playing the game wrongly. And the child was perfect in his mother's eyes and could do no wrong, so therefore my child had to be the one who was wrong in some way. The only thing I regret, looking back, was being overly harsh to my child in response because I was embarrassed and in my inexperience assumed my child was the one at fault. He wasn't, they were just different children. And strangely enough, my friend went on to have a second child who is very much like DC1, and I've had DC2 who is a well behaved little angel around strangers, and this is something we've talked about since and we are now both of us wiser parents.
And we're both a bit sad because the DC1s of this world seem to have an easier time at school and with friendship than the DC2s. So we've both had to do some supporting and building of confidence, her for her first child (who struggled hugely when the second child arrived) and me for my quiet second child.
I felt sad the other day because we met a little DC1 in the playground, a lovely outgoing confident little girl. She took a shine to my little one and wanted to play with her, but DC2 just wanted to play with me as the other child was a bit too much for her. DC1 would have been straight in there, thick as thieves... she was exactly the sort of child he adored to play with when he was little.