If both parents live with the child then only one parent can claim the child and all the benefits - they cannot be split in any way and the child should not be on the other parents tax return at all.
Only if one parent has not lived with the child at all during the last 6 months of the year can benefits be split as follows:
The parent that physically lived with the child for more than half the year is the custodial parent for tax purposes and can claim the EIC, Child Care Credit and possible file as Head of Household of otherwise qualified.
The other parent that did not live with the child is the non-custodial parent and can only claim the child's exemption and child tax credit, but ONLY if the custodial parent has released the exemption with a signed 8332 form that the non-custodial parent files with their tax return.