They can't claim the same benefits for the child. But they may be able to split the benefits. Here is how it works:
For divorced or separated parents or parents who live apart, the custodial parent, if eligible, or other eligible person who the child lived with for more than half the year, can claim head of household filing status, the credit for child and dependent care expenses, the exclusion for dependent care benefits, and the earned income credit.
The non-custodial parent, if allowed by divorce decree or consent of the custodial parent on form 8332 or similar signed statement, can claim the dependency exemption and child tax credit. For post-2008 divorce decrees or agreements, form 8332 or similar signed statement is required. The child tax benefits cannot be split any other way.
The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. Even if a divorce decree, dated after 2008, gives the non-custodial parent the right to claim the child, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form. <a rel="nofollow" target="_blank" href="https://www.irs.gov/pub/irs-pdf/f8332.pdf">https://www.irs.gov/pub/irs-pdf/f8332.pdf</a>
The special rule, for separated parents, described above, for "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons.