How do I beat the regular pirates in sword fights?

The trick is finding the proper responses to insults.  If you insult them and
they don't know the proper response, you go forward a step.  If they do know
the answer, they go forward a step and get to make the next insult.  The same
thing happens when the other pirate is doing the insulting.

I recommend that you try out a new insult the first chance you get.  Even try
coming back immediately after the current fight and use the new insult then. 
(Well, upgrade your cannons at Puerto Pollo first to empty your hold, just in
case you lose.)  If that pirate doesn't know the proper response, the next
one you fight might.  Keep trying pirates until you find one that does know
how to respond, since you're likely to get (and need) the response later on.

You might also want to throw a round or two (miss a response or give an
insult you know the other pirate can answer), since that gives you a chance
to learn new insults.