If the children live in the same home with you either custodial parent is eligible for all the benefits for one or all of the children. If you don’t file a married/joint tax return you need to decide which parent claims which child or if one parent claims all children. This is usually where divorced or separated parents seem to run into trouble by a failure to effectively communicate who claims which child on their individual tax returns. I am of the opinion that the best way to communicate with the other parent as well as the IRS in these regards is via IRS Form 8332, Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent.
If both parents happen to claim the same child usually the person who files their tax forms first will have the return processed and the person who files second will have their tax return either recalculated without consideration for the child as a dependent or maybe even have the tax return considered for examination.
The last point of significance is that only one parent is eligible to file as head of household and that parent must claim at least one of the children to use this status.