The way it works most of the time depends on the agreement between the parents such as a divorce agreement or a court order. If there is no such thing in place it is my opinion that whoever has main custody of the child should get to carry them on their taxes. Because even though child support maybe paid, the person who has the child living with them has way more of a financial burden then the child support payer.
When I got divorce, my ex husband always got to claim my son on his taxes. He always had a better job, and since he paid child support, they always let him claim my son.
I think that the person that pays over $10,000 a year should have the right to carry the child such as my husband is doing because she is using that money for rent & on herself & her other child.
I was divorced when my children were in Middle school. Their dad paid child support until they were 18 or until they graduated from High School. I claimed one son, he claimed the other. This was written in our settlement agreement and it worked out well for us. I received a very good payment for my sons, so I agreed to this. For what it's worth, the money I received did go for my sons, not me. If anything came up, such as a prom, musical instrument or car, we went half and half. The kids knew of this and I know it made them feel okay. They were old enough to understand why and what a relief for his dad and me!
I believe it should be the person paying child support.
I think the best way is to every year switch wherein, you claim them one year and your ex claims them in the next. That worked out the best for me. That way everything is 50/50 and you can't use it against each other later!