Anyone who has the unique account and unique hash number (example exam_ple#1234). Only then will other users be able to see the account. On smartphones, users can opt into syncing their contacts on their device as well as allow others on phones to find them via their phone number & email address.
However, your child’s username and profile picture may be available for everyone to see, such as by those in the same server. But, to be able to message a user, you must either be friends or in a server with them. There is also no way of seeing what servers each user is a part of.