I would suggest following her lead and just give her time. Give her the opportunity to experience new situations but don't push or force her to do anything that makes her upset. Let her come to you for comfort and if she doesn't want people she doesn't know well to hold her then don't make her (I wouldn't like strangers holding me either!). I wouldn't say she's insecure as such, I think it just shows that she has a very secure attachment to you, which is a good thing.

Yes, separation anxiety is common at around age 1 so it may just be a stage she is going through however it could simply be her personality. My 2 year old DS showed signs of being a sensitive little thing from around 3 months or so. He prefers that people don't get up close to him and he will sit back and watch in a new situation. It took a very long time before he was comfortable in mothers and play groups. He still prefers to play by himself and gets very nervous if other kids suddenly run up to him. I let him do whatever he needed to feel comfortable and if that meant sitting on my lap the whole time then that was fine. From about 18 months onwards he started exploring a lot more, has needed my reassurance a little less, and is slowly becoming a little more confident. He still cries when the GP gets too close, looks down at the ground when a teacher talks to him, and needs help in new situations, but that's okay. I know he'll learn to cope with those things in his own time and IMO the best thing I can do is to be there for him and provide encouragement and reassurance.

HTH.