There is so much missing from your question.
If you file a joint return, you are "one taxpayer" and you list all your combined income, credits, and dependents on that one return. Joint filing is allowed even if one spouse doesn't work, and it generally has lower tax rates and better deductions.
If you live together but file separately for other financial reasons, then either parent (but not both) can claim any dependent that lives with them.
If you are separated, and plan to file separate returns (married filing separate or head of household), then the first question is, where did the children physically live more than half the nights of the year? If the children lived with you, but your (separated) spouse wants to claim them, that is allowable but you need to sign a release form.
More detail about your situation is needed to give a proper answer.
