Distinct, no. Pirate speech, like the speech of maritime communities of that period, was multilingual. English, Spanish, French, Portuguese, Maritime Pidgin English, and Kreyol were all in use as needed.

Stereotypical pirate speech tics are modern inventions typically attributed to films of the 1930s. Actor Robert Newton's performance of Long John Silver in the 1950s proved a popular model. Rafael Sabatini's Captain Blood, published in 1922, depicted a pirate with rhotic Irish speech. Many 18th century pirates did spend time in and around Dublin, but not to the extent that it would influence the speech of such multiethnic pirate crews. Both pirates and royal navy were also in the habit of pressing men into service from a wide range of backgrounds, and since generations are not being bred aboard ship, there would be no chance for leveling to take place to the point where everyone shares a speech style.

J.L. Dillard, a British linguist, has this to say about the situation (as quoted in this article):

The extensive association among sailors and the African peoples they transported, worked with, and were in casual contact with, brought with it specific “exotic” or “savage” forms of transatlantic language. Cultural and linguistic contact took place in and between such cosmopolitan polyglot harbors as London, Bridgeton, Kingston, Charleston, New Orleans, Boston, Philadelphia, and coastal African factory settlements such as Bonny, Whydah, Cabinda, and Goree. A present-day cultural historian describes the linguistic situation in early eighteenth-century Jamaica as one “not of diglossia, triglossia, or even heteroglossia but of panglossia, a state of ‘generalized multilingualism,’” from which Creole emerges as the primary, if despised, language of the island. Communication in the pan- Atlantic world did not take place in standard English (or any European language), but rather in pidgins and Creoles concocted out of European and African languages. J. L. Dillard has documented how “the Maritime Pidgin English, transmitted to West Africans in the slave trade and heavily influenced by West African languages, became the English Creole of the plantations from Nova Scotia to Surinam” (All-American English, 3–76; Black English 73–185). Sailors, slaves, and those populations of whites who owned and were raised by slaves (from whom they learned this language that set them off, to no advantage, from the English) all spoke this maritime English. Sailors, naturally, were among the largest groups of speakers.

Alternate source (quoted in LanguageLog)

The speech of pirates was therefore not dialectally distinct from that of the navy of their countrymen, because of the similar make up of the crew. Individual ships may have developed their own jargon, but these would be small lexical changes only.