It depends on the type of surrogacy chosen. With traditional surrogacy, the surrogate gets inseminated by IUI. If the sperm meets the egg, she will become pregnant and be the biological mother of the child born as a result.
Conversely, with gestational surrogacy, the embryo created with the gametes of either the intended parents or donors is transferred to the uterus of the GC, so there is no possible way that she can be the biological mother of the child. If her eggs are not used, she cannot share her DNA with the baby.
In fact, traditional surrogacy arrangements are banned in all 50 US states due to the number of issues raised by previous cases, being the Baby M case the most popular one. Also, to reduce the emotional attachment that the GC could develop with the baby if the is genetically linked to the baby she's been carrying for 9 months.
Suggested for you: What Is Surrogacy & How Does It Work? – Everything You Should Know.