British English has many phrases for keeping a polite distance, and for closing it too fast.
To keep oneself to oneself: To be friendly but private, and to share little about one’s life.
A private person: Someone who does not talk about home or feelings at work.
Over-familiar: Too friendly, too soon.
Too much information: Said, often as TMI, when someone shares a personal detail that nobody wanted to hear.
To warm to someone: To start to like someone, slowly.
To break the ice: To make the first small, easy contact with a stranger.