The word "sharer" is an informal way of describing "one who shares". My niece is a sharer, she is always very generous with her toys and she doesn't mind if other children play with them. One day she will meet someone who is an even BIGGER sharer. The phrase "bigger sharer" means that she will be even MORE generous than my niece. If I see a room full of children I may be able to see who is the BIGGEST sharer, and this would be the child who is the MOST generous in the entire room.
When someone says "she wasn't the biggest sharer in the world", they are being a little bit sarcastic. What they mean is..."listen, this person is not very generous". With a child you may say that they are not generous with their toys, but for an adult they may not be generous with non-material things, like information or advice.
"She played things close to the vest" is a reference to poker, or some other game of cards. When you don't want other people to see your cards you will literally "hold them close" to your body so nobody can peek at them. When a person "holds things close to the vest" it means they are not sharing information to you.
"Is Steven going to ask Susan to marry him?"
"I don't know, he won't tell me his plans."
"I guess he's holding things close to his vest until he knows she'll say yes."
"Yeah, I think that's it."