If me and my wife are separated and we have an agreement to each claim one of our children each do I still need to put my daughter on my taxes?

Yes, if your daughter lives with you. No, if you are the non-custodial parent.

 There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. The tax benefits may not be split in any other manner.

This "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. In this case, you would not list your daughter on your return. You would each claim only one child.

Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the dependency to him.

 So, it's good idea to let the other parent know that you will be claiming those items, as many first time divorced parents are not aware of this rule and may try to claim those items, which will cause the IRS to send out letters.

Ref: https://www.irs.gov/publications/p17#en_US_2017_publink1000170897

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You can if you are the custodial parent.  The custodial  parent is (the parent the child lived with for more than 183 days in 2019.

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For tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS goes by physical custody, not legal custody. 

It is allowed  for you to arrange the children's schedules so that one child spends more than half the year with the father while the other spends more than half with the mother. Then you are each the custodial parent of one child.

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 IRS goes by physical custody, not legal custody. 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, or you have a verbal agreement with the other parent, he must still get form 8332 from the custodial parent. A properly worded decree should require her to provide that form.