If you are going abroad without the other child’s parent, there are definitely things you need to be aware of.
In particular, with child trafficking and kidnapping being prominent, border officials want to know that you have parental authority to remove the child from their country of residency.
As a general rule, you’ll want a signed/notarized letter from the other parent authorizing their international trip with you.
There are exceptions to this, of course, in relation to custody, etc.
You can and should review these requirements on the US State Department website, but I also recommend this attorney’s more easy read discussion on the matter.