There are a lot of questions that need to be asked...

She is self-employed as in running a business as a sole proprietorship?

What is the type of legal entity that business is registered as in Denmark?

Does she have a limited liability for this business?

Did that US company enter into a contract with her business entity or her as an individual?

What type of services did she render?

Are there terms in the contract that dictate where the services were to be performed?

Where did she perform those services in relation to that contract?

You said "[t]hey filed their taxes in Denmark, and paid taxes in Denmark" but is she paying income tax to Denmark on that income she received from this US company?

Is she paying social security to Denmark (given territorial rule would apply based on the limited facts you provided)?

Is she a US citizen?  If not, is she a green card holder?