In this question we learned that pirates did not really talk like they are commonly portrayed. 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:
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:
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.