The US government places no restrictions or requirements on you when exiting or entering the country with your child. Many foreign governments might require require documentation for entry or international air carriers to board their plane.
In the USA, though, unless there's a court order requiring permission from both parents, a child may travel (even to overseas) with either parent/guardian without showing proof of permission from the other parent.
Is a minor travel consent form required for domestic travel? No, a minor travel consent form is not required for domestic travel within the U.S. However, it is recommended that parents and/or legal guardians carry a notarized letter of consent when traveling with a child.
Most U.S. airlines will permit children who have reached their fifth birthday to travel unaccompanied. Kids ages 5 through 11 who are flying alone must usually travel pursuant to special “unaccompanied minor” procedures. On some airlines, these procedures are required for unaccompanied children as old as 14.
The letter of consent, preferably in English and notarized, from the other parent (if the child is traveling with one custodial parent) or signed by both parents (if the child is traveling with a guardian or alone) should say: "I acknowledge that my child is traveling outside the country with the name of the adult ...
A consent letter is recommended for all cross-border travel, even for a day trip, if the child will be travelling alone, with only one parent, with friends or relatives or with a youth group.
If the child is accompanied by only one parent, the parent should have a note from the child's other parent. For example, "I acknowledge that my wife/ husband is traveling out of the country with my son/ daughter. He/She/ has my permission to do so."
You should obtain a child travel consent letter granting you permission from the absent parent to leave the country with your child (having it notarized is not required, but highly recommended).
The letter should say: "I acknowledge that my child is traveling outside the country with the name of the adult with my permission." A parent who frequently crosses the border by land with a minor should always carry a letter of permission from the other parent.