If yes, do they bear any resemblance at all (whether in lyrics or in melody) to later ditties that we popularly associate with pirates of the era, like “Fifteen men on a dead man’s chest” from Treasure Island or “A Pirate’s Life for Me” from Disney’s Pirates of the Caribbean ride?
I have done some research in the past on pirate songs. To summon the answer up; It turns out that pirates at that time had little interest in writing down their songs on music sheets.
Most of what we can consider authentic sea shanties come from the 19th and early-20th century, and often from the navy. Many other now famous songs actually come from the mid 20th century and don’t have any origin with sailors and sea; just songwriters who romanticized the sea.
It’s hard to find original sources for any information regarding these songs and where they came from. Most songs were spread by word of mouth, and thus we have little information on the origin of many songs. And since nobody recorded their songs (since recording wasn’t invented yet) we have no actual way of knowing how those songs would have sounded.
There is the book *The Complaynt of Scotland* from 1549 which apparently holds some old Scottish ballads and songs, such as *The Frog Went A-Courtin*. I don’t know if there are any accurate reproductions of the songs in the book, but it’s the oldest relevant source I can think of.
I, myself, am Dutch. I have had schooling in Dutch history (not academic, and a lot autodidact), so I will focus on two Dutch songs. The *Wilhelmus* (National anthem) and *Al die willen te kaap’ren varen*.
The Wilhelmus is the Dutch national anthem and the oldest national anthem in the world, with the melody coming from the French song *O la folle entreprise du prince de Condé* from 1568, and the first lyrics being written down in 1574. What makes this song a pirate song is that is was sung by the Geuzen (English: Beggars), which were the eventually victorious rebels during the Dutch Rebellion against the Spanish rule (1568 – 1648). Before the Geuzen managed to get a foothold on Dutch land, they lived primarily as pirates or privateers. It was a popular song for at least 200 years until being forgotten somewhat. During these 200 years the Dutch were infamous for their adventures on the sea (Dutch Golden Age), often claimed to be piracy or privateering. 
Al die willen te kaap’ren varen is a Dutch song whose lyrics have a very obvious connection to the sea. The song itself describes how a good sailor should be (have beards, like old fish, like hardtack, etc). There are two main reasons why we could link this to a pirate song. The first of all is the oldest written source from 1903: Het oude Nederlandsche lied by Florimond van Duyse. This book mentions an even older source: Chants populaires des Flamands de France (1856) by E. de Coussemaker. Both books describe the songs as being a lot older than the moment the books were written. It’s not unlikely that this song dates back to the Golden Age of Piracy. The second reason is that the main course of the song is about “Kaap’ren varen” which could be interpreted as either “kaap varen” (privateering or piracy) or “naar de Kaap varen” (sailing to the Cape of Good Hope, an important place during the age of piracy). This could also be relative to the Dunkirk Privateers, as de Coussemaker could have learned the origins of the song in Dunkirk.
EDIT: As for how these songs originally sounded:
Kaap’ren Varen follows the tune also sung [here](https://www.youtube.com/watch?v=hBeJOOxfOOA), although the backing music may not be as original as the tune sung here.
1. (Dutch) https://www.koninklijkhuis.nl/onderwerpen/volkslied/
2. (Dutch) Nederlandtsche Gedenck-Clanck – Adriaen Valerius
3. (English) Bandits at Sea: A Pirates Reader – C. R. Pennell
4. (English) English/British Naval History to 1815: A Guide to the Literature – Eugene L. Rasor
rahl422000: Related question, are the “sea shanty’s” from Assassins Creed games based on actual songs?
As correctly pointed out, most of the songs we now associate with “pirates” are creations of later eras — which is true of quite a lot of the culture and errata surrounding pirates as a rule.
To add to his answer, “Dead Man’s Chest,” which you inquire about, was an entirely fictional creation of Robert Louis Stevenson. He made it up (without any music or description of a tune) just for the novel *Treasure Island* in 1883. It was never sung by any pirate crew. “Yo Ho (A Pirate’s Life for Me)” was written by two Disney songwriters (George Bruns, who wrote many famous Disney numbers, and Xavier Atencio) in 1967 as a kind of tribute to what was seen as pirate-y songs, rather than an attempt at a real sea shanty.
To understand why sea shanties (that is, what we think of as “pirate songs” today) were unlikely to actually have been sung by pirates or by anyone during the so-called Golden Age of Piracy, it’s worth understanding their genesis. Sea shanties took their inspiration from land-based work and military songs, most especially from those of African-Americans, including slaves, in the antebellum period, which accounts for their rhythmic, work-friendly nature and their call-and-response structure. For example, a common, real-world sea shanty is [“Santy Anno”](https://en.wikipedia.org/wiki/Santianna), which originated around the time of the Mexican War, and was sung with different lyrics by the British/Canadian (pro-Santa Anna) and American (anti-Santa Anna) merchant navies, respectively. This shanty features the classic elements: military cadence, call-and-response, and a rolling, continuous structure to allow it to be used when a crew was endlessly turning a capstan (or working a pump). It would not have been well suited to the kinds of hawser-hauling, rigging-climbing work done on a square-rigged bark, for example, and clearly features inspirations from cultural and musical features that pirates were unlikely to have run across in the 16th-17th century.
Now, pirates would certainly have known songs and been apt to sing them, perhaps together — though this was more likely to occur on land, in taverns, than out at sea. If they were to sing songs, they might sing popular folk songs of the era, such as “Barbara Allen” (https://en.wikipedia.org/wiki/Barbara_Allen_(song)) or [“Spanish Ladies,”](https://en.wikipedia.org/wiki/Spanish_Ladies) if they were British.
* *Songs of the Sailor and Lumberman*, Dorflinger, William (1990)
NB: I still have some MP3s I downloaded via Napster and Kazaa. Mostly misnamed 80s and 90s pop hits.