How do the children get a copy of the power of attorney that our aunt provided to the hospital?

Why doesn't your aunt provide you with a copy? While that isn't a requirement, I don't understand why this information isn't widely shared. Parties that have to act on it, such as a hospital respecting the decisions of the medical POA, must see a copy. But I would think family members and perhaps close friends should know what to expect and who to trust with decisions.

But I agree with GA. The hospital does not have authority to release copies of this document.

If your aunt wanted them to have a copy, presumably she would have given it to them. The hospital staff acted properly; that kind of information is confidential and it would be inappropriate to provide it to family, especially since a hospital is not a party but rather an end provider. I. e., absent specific enabling provisions, the hospital wouldn't have the authority to provide anyone else with a copy.