A qualifying child for the earned income tax credit (EITC) must meet three tests: age, relationship and residency. Your son or daughter or lineal descendant of your son or daughter passes the first two tests if he/she is either under age 19 or under age 24 and a full-time student. The qualifying child must also reside with you in your home for more than six months of the year. Temporary absences for illness or school are OK.
It is possible, under the living conditions you describe, that the child is the qualifying child of both the parent and the grandparent. In such cases, either the parent or the grandparent can treat that child as their qualifying child as long as the grandparent has a higher adjusted gross income (AGI) than the parent. If the parent's AGI is higher, then only the parent may treat the child as a qualifying child. Once the determination is made as to who may treat the child as a qualifying child, it is that person who can claim the EITC, assuming that all the other rules for the EITC are met.
Also, please note that once it is determined which taxpayer has the qualifying child, the other taxpayer is not entitled to any other tax benefit for that child.