No, you do not issue a 1099-MISC. Form 1099-MISC is only for payments you make in the course of your business. It is not for personal payments.
Some people mistakenly think that they need a 1099-MISC in order to report the income on their own tax return. They do not. They can and should report it based on their own records of how much they were paid. Apparently your caregiver is one of the people who has this misunderstanding.
