The main distinction in an adoption is whether it is open or closed. This refers to whether the birth parents and the adoptive parents have any contact with or knowledge about each other. The Texas Department of Family and Protective Services handle most closed adoptions in the state, and the adoption is completely anonymous with no contact between the birth parents and the adoptive parents and child. In an open adoption, the adoptive parents raise the child, but the birth parents can visit and spend time with the child.
A private adoption refers to an adoption where the parent or legal guardian places the child with a new family personally. It is illegal for anybody to buy a child in the state of Texas, so the adoptive parents should never pay any money directly to the family. Similarly, it is illegal to pay an attorney for the placement of a child. The adoptive parents can pay medical bills directly to the hospitals and reasonable attorney fees for representation throughout the adoption process, though.
International and interstate adoptions can be very complicated because the adoptive parents must obey all adoptions laws set forth both in the child's current country or state of residence and their own. In general, it is important that the adoptive parents not move the child to their home until both the sending and receiving state or country approves of the adoption. With international adoptions, the parents can even receive a Texas birth certificate for the child.
With any adoption, an attorney's advice and representation can be very valuable to the adoptive parents so that they can feel confident that they completed the process properly and can start to make decisions for the child.
Source: State Bar of Texas, "Adoption Options", October 11, 2014