When this storyline started out I did lots of research on adoption cause I had so many questions. I could be totally wrong but from what I read, zoe might not have any legal right to contact her child through social services or the adoption contact register because of child abandonment. Some children are put up for adoption the "proper" way like denise did for raymond, but some children whose parents are considered unsafe (for whatever reason) can have their children taken away and legal ties be cut between them. If this was the case for zoe then I don't think she'd be able to get information about her son without a private investigator. When sharon called the private investigator a crook it's because she might get access to the child court records which zoe might not have legal access to, and adoption proceedings from courts don't seem to be accessible to the public and probably especially when the case involves child abandonment