Just to be clear, no they didn't. However, what they do use were privateers. At least that seems to be how they thought about it, Fernand Braudel suggest it wasn't until the 17th century that it was really named as piracy. That piracy was something Christians honourlessly did to other Christians in the Atlantic. He does note this distinction may seem strange to the reader and goes on to discuss how "privateering" essentially had a semi-formal structure and rules that were often adhered to, even though out-right letters of marque were uncommon (if I read it right only used against other Christian targets). Not in any way any chivalrous ideal, just that accommodation, discussions and negotiations occurred and that extensive networks crossed the Mediterranean to facilitate return of prisoners, sales of goods etc. And that this enterprise had long traditions predating most Western European powers and the Ottomans. The interesting thing is it wasn't really seen as piracy at the time apparently.

Braudel basically describes it as everyone's war on everybody in the Mediterranean. The first motive was profit, secondary came considerations of who or what was captured. Uskoks in Fiure and Segna raided Christians and Muslims both. Western ships would attack Venetian and French shipping to inspect for "jewish and turkish goods" which they would then expropriate as "contraband" and neither Venice's or the Pope's complaints were considered. Ottoman galleys likewise inspected ships for Neapolitan or Sicilian goods.

And how could you tell who the attackers really were? Braudel describes an incident in 1536 when Ibiza was raided. By the French or "Turks"? He blames the French because the raiders took substantial amounts of salted pork with them... He continues to describe many incidents where the perpetrators were co-religionist. And the English when they showed up were apparently so predatory everyone complained. That said Braudel also says something about the difficulty in establishing who did what. Many of the "Barbary corsairs" were people born in northern Protestant nations out to make a buck or two.

In 1563 Sancho de Leyva travelled to the Barbary coast with Sicilian galleys to capture rowers. In 1576 the marquess of Santa Cruz "patrolled" the coast of Tunis. As Braudel puts it "other would have called it a piratical raid against the impoverished Kerkenna islands". And just as a fun twist the Sultan would often complain to Venice they did not do enough to curtail the incursions of western corsairs... talk about having to walk a thin line of diplomacy.

Broadly speaking with the loss of Rhodos in 1522 the Western "piracy" was mainly concentrated to the western parts of the Mediterranean, and between 1538-1571 the Ottomans were largely masters of the sea, but after Lepanto broke the dominance of the Ottoman fleet the eastern parts were raided with a vengeance by Christian privateers (pirates the victims likely would have said). It should be noted we think of it in terms of national privateers, but it existed on all levels, nations, principalities, cities (cities were one of the big players as they had finance and legal wriggle room), villages and even individuals. Not all privateers earned big money some were small opportunist of a dozen people and a small boat.

Like I say from Braudel's description of the economy of the Mediterranean and the business of piracy therein the best description is probably as he put it an everyone's war on everybody.

Ferdnand Braudel, La méditerranée et le monde méditerranén à l'epoque de Philippe II, 1991, (the 9th edition, and I got a Swedish translation of it from 1997)