You may wish to look up RH incompatibility.
As a woman who has Rh negative blood, I was told that if my baby's father had Rh positive blood, then I would need a special shot at the birth of the baby to prevent my body from producing antibodies to the Rh differential that could happen if any of the baby's blood came in contact with my body while being born and which would cause my body to react against any future pregnancy. The mother and baby's blood coming in contact only happens incidentally - they are not "mixed" and the baby does not "get" its blood from the mother. It is entirely its own.
According to the Mayo Clinic (http://www.mayoclinic.com/health/rh-factor/MY01163/DSECTION=why-its-done):
Typically, the antibodies aren't a problem during the first pregnancy. The concern is with a subsequent pregnancy with an Rh positive baby. In this case, your existing Rh antibodies might cross the placenta and fight the baby's red blood cells. This could lead to life-threatening anemia — a lack of healthy red blood cells — for the Rh positive baby.
If you're Rh negative, you might need to have another blood test — an antibody screen — during your first trimester and again during week 28 of pregnancy. The antibody screen is used to detect antibodies to Rh positive blood.
If you haven't started to produce Rh antibodies, you'll need an injection of a blood product called Rh immune globulin. The immune globulin prevents your body from producing Rh antibodies during your pregnancy.
If your baby is born Rh negative, no additional treatment is needed.