Can I choose a birthparent and adopt their baby or does the agency choose the birthparent?
In open adoption the birthparent chooses the adoptive parents. The adoptive parents then decide if this is a good match for them. If not, they can decide to unmatch with the birthparents.
In closed adoptions the social worker or attorney acts as a go-between for the birthparents and adoptive parents, who never meet one another. While this type of arrangement may seem beneficial to adoptive parents, it turns out to be less healthy for all parties involved. For more on this see benefits of open adoption.