Map notes are not meant for simultaneous editing. What I suspect is happening is that as each child saves their edit, it overwrites any existing edits. I suppose if the children made their edits sequentially, you could avoid the overwrite problem. Even with unique usernames, you won't be able to achieve what you want with map notes because map notes are stored in the web map that contains them and only the owner of the web map can make changes to it.
The only way to achieve what you want is to create a hosted feature service, which requires a subscription to ArcGIS Online.
Thanks,
Mike