There is broad consensus that there was contact with South America; the primary evidence being the extensive cultivation of the sweet potato (or Kumara, Kumala, etc) in the Polynesian Pacific which was originally domesticated in Central/South America. Furthermore there is possible chicken DNA crossing, human genetic evidence, and the presence of the coconut in South America which may indicate more extensive contact than just one instance of voyaging- yet, that further evidence is subject to debate.

As to the other islands; the pattern is usually one of exploration from a central island; for example from Tahiti or the Tuamotus northward along the winds to discover Hawaii- a voyage that may have taken place intentionally as part of a project of exploration, or accidentally as a fishing or voyaging expedition between known sites gone wrong. This is usually followed by voyages of settlement; we know the voyages were intentional, otherwise they would not have brought stores of seeds and pairs of animals to reproduce. This could have been on voyage- or more likely several as many of the Pacific Islands required some amount of terraforming to make the islands more habitable for people. Many of the islands had little in the way of food for people until it was planted and could be harvested.

After these initial voyages of exploration and settlement, the island chains would often stay in contact with one another. Both Hawaiian and Tahitian oral traditions discuss episodes of voyaging between the island chains as families remained in contact and continued to exchange goods and people for several generations. It seems gradually voyaging grew less and less common across long distances; the islands simply didn't have lots of resources that were not also present at home, which reduced the impetus for the long voyages.

So that little overview mostly covers long distance voyaging. As to inter island voyaging- amongst islands of an archipelago like Tahiti or Hawaii; voyaging was extensive and continued constantly. It was relatively common between island chains that were near each other as well, such as Tahiti and the Tuamotus. Or, extremely regular, such as the voyaging that took place between Samoa-Tonga-Fiji.

Knowledge was pretty well known amongst those in the position of navigator or priest. Tupaia is perhaps the best example of this in the past; he was a Tahitian priest/navigator who sailed with Captain Cook for a while and at one point drew a European style map for Cook- attempting to translate the geographical-voyaging knowledge he had gained from oral traditions to the representational style used by Europeans to accomplish overlapping goals. He drew out some forty islands that have since been placed.
Many oral traditions contain geographic reference points which has been another way in which historians or anthropologists can ascertain the extent of geographical knowledge or memory.

So, how much did a commoner in Tahiti know about Hawaii? That would likely depend on if they had a genealogical connection to someone who left as a settler to Hawaii or were charged with maintaining the oral histories or traditions that contained reference to the distant places.

Some academic reading if you want to explore the topic in some more depth:

• D’Arcy, Paul. People of the Sea: Environment, Identity, and History in Oceania. Honolulu: University of Hawaiʻi Press, 2006.

D'Arcy's work is a very recent and well regarded; he discusses voyaging to a good extent, especially in Micronesia, where the voyaging traditions maintained themselves up until the present and have since spread back out across the Pacific and help inform the work of the Polynesian Voyaging Society and others.

• Irwin, Geoffrey. The Prehistorical Exploration and Colonisation of the Pacific. Cambridge: Cambridge University Press, 1994.

Irwin uses lots of computer simulation and explores how they work and what they show us. His focus is really on the initial stage of exploration and how Polynesians- and other Oceanic navigators were able to accomplish such an extensive settlement of the Pacific.

• Kirch, Patrick Vinton. On the Road of the Winds. Berkeley: University of California Press, 2002.

Great overview; very thorough, and covers everything from settlement to continued trading and the likely voyages to South America.

The ancient polynesians were skilled navigators, having crossed thousands of miles of open ocean on outrigger canoes. They used a combination of navigation technquies including using the stars, observing ocean/wave patterns and flights of birds. Most of their knowledge was through oral tradition, having no written language. The knowledge was passed down from generation to generation through song and chant.

About the Polynesians making contact with South America, I wrote a short paper about it last semester for my history class so let me just copy and paste a paragraph here:

There is evidence to suggest polynesians could have gotten as far as south america. The sweet potato which is native to south america, was already widespread in polynesia when European explorers observed it in there in the 18th century. [1] Furthermore the Quechua (South American native language) word for sweet potato is k’umara. Easter Island word for it is kumara, Maori word for it is kumāra. [2] Is this just a big coincidence or could have they learned the word from the South American natives?

Also a chicken bone in Chile was found that was radiocarbon dated to be from before European contact. The chicken bone also showed genetic similarities to chickens from Tonga and Samoa. This suggests that the chicken was introduced to Chile via the Polynesians. [3]

There is also genetic evidence; two studies suggest that the Easter Island population have some American Indian genetics that are pre-European contact. I also read that on Mapuche Island off the coast of Chile, human bones found have “Polynesian features” [4].

back to your original question, there is evidence that Polynesians made continuous deliberate voyages back and forth throughout the Pacific. Adze tools were found on Napuka Island in French Polynesia are made of rocks that originate from Pitcairn island, Marquesas island, Rurutu island and most notably Kahoolawe island in Hawaii [5] This suggests that there was travel between these islands in French Polynesia and Hawaii. Also in 2009 Hawaiian-style petroglyphs were found in Tonga [6].

the only way for these things to happen were for the polynesians to be expert navigators capable of travelling over long distances at will and expert craftsmen having the ability to build their outrigger canoes to make the voyage.

[1] Gibson, Arthur C. "BATATAS, NOT POTATOES." CLA, 2011. Web. [2]"Genetic relations of South American Indian languages". In Adelaar & Muysken, eds, The Languages of the Andes. Cambridge University Press, 2004 p. 41. [3] Storey, A. A., J. M. Ramirez, D. Quiroz, D. V. Burley, D. J. Addison, R. Walter, A. J. Anderson, T. L. Hunt, J. S. Athens, L. Huynen, and E. A. Matisoo-Smith. "Radiocarbon and DNA Evidence for a Pre-Columbian Introduction of Polynesian Chickens to Chile." [4] Lawler, A. "Beyond Kon-Tiki: Did Polynesians Sail to South America?" Science 328.5984 (2010): 1344-347. Web. [5]Collerson, K. D., and M. I. Weisler. "Stone Adze Compositions and the Extent of Ancient Polynesian Voyaging and Trade." Science 317.5846 (2007): 1907-911. Web. [6]Egan, Shane, and David V. Burley. "209 TRIANGULAR MEN ON ONE VERY LONG VOYAGE: THE CONTEXT AND IMPLICATIONS OF A HAWAIIAN- STYLE PETROGLYPH SITE IN THE POLYNESIAN KINGDOM OF TONGA." The Journal of the Polynesian Society 118.3 (2009): n. pag. Web.