The formal statement of the warrant was to authorize the agent to pass beyond the borders of the nation ("marque", meaning frontier), and there to search, seize, or destroy assets or personnel of the hostile foreign party ("reprisal"), not necessarily a nation, to a degree and in a way that was proportional to the original offense. It was considered a retaliatory measure short of a full declaration of war, and by maintaining a rough proportionality, was intended to justify the action to other nations, who might otherwise consider it an act of war or piracy. As with a domestic search, arrest, seizure, or death warrant, to be considered lawful it had to have a certain degree of specificity, to insure that the agent did not exceed his authority and the intent of the issuing authority.
The United States Constitution (Art 1 sec. 8) authorized only Congress to issue letters of marque and reprisal. Such issuance is not restricted to private actors, however, and such a warrant could also be issued to the President, as an authorization for limited offensive warlike operations outside the territory of the United States.
The difference between a privateer and a pirate was a subtle (often invisible) one, and the issuance of letters of marque and reprisal to private parties was banned for the signatories of the Declaration of Paris in 1856. The United States was not a signatory and is not bound by that Declaration, but did issue statements during its internal 1861-65 Civil War, and during its 1898 war against Spain, that it would abide by the principles of the Declaration of Paris for the duration of the hostilities.