John Paul Jones was a privateer.Just check this info from: http://www.history.navy.mil/faqs/faq58-1.htm
"In December 1775 John Paul Jones received his lieutenant's commission from the Continental Congress for its navy."
"Most general histories of the Navy focus on his seamanship and courage when in danger but not on his character. These honorable portrayals of Jones contrast with images of piracy presented by the British. Rudyard Kipling, for example, refers to the "exploits" of Jones, "an American Pirate." Sir Winston Churchill calls him a "privateer" and even Theodore Roosevelt mentions him as a "daring corsair." Jones, of course, neither held a privateering commission nor was engaged in piracy, the realization of which might be what prompted someone on one occasion to cross out the British Library catalogue entry for John Paul Jones as "the Pirate," and substitute "Admiral in the Russian Navy." In modern terms John Paul Jones indulged in questionable behavior, since his popularity with women led him to having many lovers. Furthermore, he failed to be a good team player, spurning the naval efforts of others as inadequate compared with his own brilliant accomplishments."
"After the Revolutionary War, Commodore John Paul Jones was active in negotiating prize money claims in Paris. In 1788 he entered the service of the Empress Catherine the Great of Russia as a rear admiral. He hoped that command of a battle fleet in Russia would qualify him for higher command if and when the United States built a permanent Navy."
He was a commissioned officer of the U.S. Navy. Any attacks on British ships were done under a flag of war.