I think for the most part couples who do this try to keep it anonymous, but it's not always possible. One reason is that the parents may want to keep a connection with the donor (through the donor agency or through a third party) because of medical history. If the donor learns that she has some kind of genetic issue, for example, she is legally and morally obligated to let the family know. Other reasons are -- the family used a relative as their donor and will be seeing that person at family events and don't want family secrets -- or they just want to be open and honest with their child, and the child chooses to discuss it with friends or other members of the family. So I think it's hard for families to keep this as a hermetically-sealed secret.
I posted because I am deeply involved in this issue and have many friends who have been through this. I have read a lot of critical or even hateful things on the internet about this, and I think about how it feels to my friends (but they are adults and can deal), and more importantly, the children. I read an article on boston.com (not sure if it's still there) about this and the hatefulness of the comments just stunned me.
I do see a stigma developing -- there's even a stigma associated with just having standard fertility treatments. This is why celebrities are often so quick to say "it wasn't IVF!!" as if there is any shame in that. Angelina Jolie, Julia Roberts, etc. I don't think it's a terrible stigma, and I think most decent people know these things are private and it's none of our business. I just don't want there to be any stigma at all, on any child. Yes, if you are a donor egg or donor sperm baby, you have genetic links to people outside your family. Be grateful to them for helping to give you life. Then always remember, your parents are the people who loved you and took care of you, end of story.
thanks for listening ~