In this question we learned that the common portrayal of pirate English is not historically accurate. Given that they were professional sailors, they probably had a wide store of nautical jargon; but what would be an example of speech that would typify a pirate, such as Blackbeard? Would the captains and officers have notably different speech from the rest of the crew?
I found the following on the Wikipedia page for International Talk Like a Pirate Day, all attributed to Blackbeard, as fate would have it:
From Lt. Robert Maynard's report of Blackbeard at the Battle of Ocracoke:
He styl'd us 'young puppies' and shouted 'May the Devil take my soul if I ever gives quarter or asks it of ye!'
Damn ye, yellow-bellied sapsuckers, I'm a better man than all of ye milksops put together.
The only written records recovered from Blackbeard's ship, the Adventure, after his death ran as follows:
Such a day, rum all out- Our company somewhat sober- A damned confusion amongst us !- Rogues a-plotting - Great talk of separation- so I looked sharp for a prize- Such a day found one with a great deal of liquor on board, so kept the company hot, damned hot, then things went well again.
Other than the use of ye and the colorful use of metaphor, nothing really stands out.
An article on Slate claims the following about actual Pirate speech:
So, was there a typical pirate accent at all? Among British outlaws, yes: The onboard speech was most likely underclass British sailor with extra curse words, augmented with a polyglot slang of French, Italian, Spanish, and Dutch picked up around the trade routes.
Pirate captains and officers seem to mostly to have been former pirates who just happened to have good (pirate) leadership skills, so most likely they would have not talked noticeably differently than the rest of the crew. Blackbeard, for instance, started off a as a crewmember on another pirate ship. His mentor in turn appears to have been a commoner as well, as was Calico Jack (creator of the skull and crossbones flag).
I've just added this answer to the question you linked to (as I think it's more appropriate there) - but to summarise, there are some good reasons why English pirates were likely to come from the south west of England, so it's feasible that the Westcountry accent used in popular portrayals was indeed prevalent.
We do know, though that many (or at least, some) privateers went on to become pirates, and many privateers were members of the aristocracy, although they probably sourced their crew in such a way as to minimise costs and maximise profits; so I think it's highly likely that there would be an audible distinction between the accents and dialects of senior officers and their crew.
The real answer is, of course, we just don't know for sure.