If she were applying to be, say, an undersecretary of education in Barack Obama’s new administration, Caroline Kennedy would have to fill out a 63-item confidential questionnaire disclosing potentially embarrassing text messages and diary entries, the immigration status of her household staff, even copies of every résumé she used in the last 10 years. If she were running for election to the Senate, Ms. Kennedy would have to file a 10-part, publicly available report disclosing her financial assets, credit card debts, mortgages, book deals and the sources of any payments greater than $5,000 in the last three years. But Ms....