If you are both strictly following the rules, there should be no need for you to speak to each other about who is claiming what regarding the children. It sounds like the two of you have just been making decisions without regard to the tax laws.
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 (183 nights in 2019), 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 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.
In tax law, there is no such thing as 50/50 custody. One parent or the other is the custodial parent who the child lived with more than half the nights of the year. The parent who can't claim either the custodial child benefits or the dependency should not even enter the child or their tax return.
