The IRS will make the determination regarding who is entitled. It is unfortunately, a long process.

If you were the first to e-file, their return will reject. If someone else claimed your child when they shouldn't have, and if they file first, your return will be rejected. The only way to claim your child is to file a return on paper. The IRS will process your return and send you your refund. Usually within a few months, you'll receive a letter from the IRS, stating that your child was claimed on another return. If you made a mistake, file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, removing  the child from their return,  the IRS, will require you to provide proof. There will be a list of possible ways you can do this. The more you can provide, the better your chances.

 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 on the divorce decree.

Below are three links that may help you get a head start on the paperwork you may need if you need to prove you are the one entitled to claim.

https://elite8financial.com/wp-content/uploads/2017/01/f886-h-eic.pdf

https://www.eitc.irs.gov/eitc/files/downloads/f886-h-hoh.pdf

https://www.eitc.irs.gov/eitc/files/downloads/f886-h-dep.pdf