But my child lived outside the United States for quite some time.
In that case, you'll need to provide a document that proves that your child lived in another country and does not have a social security number. You could use records like the child's school records or employment records.
How can I guarantee that my personal Social Security account will not be shared by the agency?
you are the only one who can create your personal my Social Security account for your exclusive use. No one can create or use an account on your behalf, even with written permission.