It's a confusing topic, complicated by difficult statutes, but let me try to clear up the question a little.
First, yes, some children "automatically" acquire citizenship. That happens everyday for kids born here in the U.S. But -- and this is where all the confusion begins -- what about kids not born in the U.S.?
There are certain conditions where a child not born in the U.S. may acquire U.S. Citizenship. Let me focus on foreign-born children who are not U.S. Citizens by blood (such as kids born to American parents stationed overseas -- that's for another article).
For this article, I am focusing on a child who was born overseas, and who somehow acquired American parents after birth. This can be under a variety of scenarios. Consider this hypothetical example: a mother, Maria, immigrates to the United States with her 1-year-old child, Silvia. Maria and Silvia both receive green cards. Maria applies for naturalization after 5 years of permanent residency. Sometime around the Silvia's 8th birthday, Maria takes her Oath of Allegiance and becomes a U.S. Citizen? What about Silvia? Read on.
Some children may acquire citizenship "automatically" -- meaning by operation of law without the need to apply. This, however, is not the same thing as declaring said citizenship. If a child qualifies, the government will not seek out the parents to inform them of the child's qualification. Parents who believe their child meets the qualification for "automatic" citizenship may file for a Certificate of Citizenship with DHS. Parents may also apply for a passport with DOS.
This automatic acquisition of U.S. Citizenship occurs when all of the following conditions are met:
1. The child has at least one U.S. Citizen parent;
2. The child has not had its 18th birthday;
3. The child resides legally in the United States in the legal and physical custody of the USC parent or parents; and
4. If adopted, the child meets all requirements for adoption under INA.
In the hypothetical scenario for Maria and Silvia, Maria became a U.S. citizen through Naturalization. As such, Silvia then qualified for "automatic" citizenship because Silvia then had a USC parent, Silvia was under the age of 18; Silvia had a green card; and Silvia lived with Maria, her mother, in the United States.
Note that it is not possible to acquire "automatic" citizenship if a child was born and resides outside of the U.S. In such cases, the child must naturalize and take the oath of citizenship.
Here is a little bit more info on children that may affect the determination of citizenship: The INA defines a child differently for purposes of naturalization. As used in Title III of the Act (regarding Citizenship and Naturalization), the term "child" means someone who has not yet married, and who is not yet 21 years of age. If adopted in the United States, the adoption must occur before the child reaches the ages of 16. INA §101(c)(1).
For purposes of Titles I and II, child is defined differently, and both definitions must be considered for the "automatic" conference of citizenship. INA §101(b)(1) defines a child as an unmarried person under the age of 21 who is any of the following:
1. born to legally-married parents;
2. a step-child who was under the age of 18 at the time of the relevant marriage;
3. legitimated under law prior to the child's 18th birthday;
4. born out of wedlock but who sought a status or benefit through his parents; or
5. adopted before the age of 16, or who is a sibling of a child adopted before age 16.