Insurance handler here - just give your name, registration, and insurance details.
Never give your full personal details, as some people are a little crazy, and I am forever dealing with complaints about people involved in the claim harrasing the other person and accusing each other of lying.
If you do provide your full details, and it's not your fault, the other person's insurers may contact you directly to offer their services - saves you claiming through your own insurance, plus there is no excess to pay.