I can’t say about the West, but there is some strong but still debated evidence about contact from the East.

The four main things (as laid out in this /r/askanthropology post by /u/GrumpySimon) are

1. sweet potatoes (linguistically and biologically),

2. the existence of sewn-plank canoes and a certain style of fish hooks, characteristic of Polynesia, and the names for them among two small groups in California,

3. possible but by no means definitive pre-Columbian chicken bones, seemingly from Polynesia rather than Europe,

4. ambiguous genetic data, the best seemingly indicating that we have contact on Rapa Nui (Easter Island) but it’s unclear if that’s people from the mainland moving East or people on the Islands moving West and back.

For more depth, see /u/Reedstilt’s and /u/Cozijo’s responses for more depth about points one, two, and three.

• Do you think it's plausible that ancient Polynesians reached the Americas? If so what do you think contact would have been like?.

The only more in-depth place I can suggest going for point four are the resources listed here and the link in /u/GrumpySimon’s post.

So there’s no written or specific oral record, but there are several things that indicate some degree of contact between Polynesia and North and South America, and not just of artifacts but of living people.

Based on the flow of ocean currents through the Atlantic, the place to look for haphazard debris from Europe would be the northern coast of South America. So far we have nothing conclusive on that front though, conspiracy theories regarding a Roman shipwreck that the Brazilian government is hiding for some inexplicable reason aside.

Now, if by "from the west" you would have meant west-of-the-Americas rather than western Eurasia, the story is a bit different. The currents of the north Pacific flow toward North America, so debris from East Asia, and Japan especially, washes up here all the time.

As an extreme example of this, there's the case of the Hojunmaru, a cargo ship that vanished in a storm off the coast of Japan in November 1832 and washed ashore in January 1834 among the Makah in what's now Washington. The Makah came aboard to salvage what they could from the wreck: iron fittings, coins, ceramics, a map of Japan, and the three surviving members of the crew. That's what makes the Hojunmaru incident interesting. Not only did material cross the Pacific, people did too, in this case, the ship's navigator Iwakichi and two cooks, Kyukichi and Otokichi.

The Makah kept the Three Kichis, as the survivors came to called, as slaves. The map they traded to their neighbors, who traded it away to their neighbors, and so on until it reach the Hudson's Bay Company at Fort Vancouver. The Company sent men to pay the Makah a ransom for the Kichis, and attempted to use them as leverage to open trade with Japan. But they three sailors ended up in China when those diplomatic efforts fell through.

I can't say about Flotsam and Jetsam, but I can say that once Europeans arrived there where some interesting examples of junk salvage by both groups.

For example, many natives started to apply stoneworking techniques to glass that settlers had left lying around at old campsites. Glass as a material is actually remarkably similar to flints and Obsidian in certain regards, the key similarity being the formation of Conchoidal fracturing, which makes them break predictably, and thus suitable for making stone tools, in this case arrowheads, blades and end scrapers.

Glasses, synthetic and natural (Obsidian), are prized by stone using socieities for the incredibly sharp edges that can be created. Flint is certainly more common and reliable, but having foreigners leave perfectly good glass lying around must have seemed like madness to the locals.

And the Europeans started salvaging the natives arrowheads that they discarded when they lost their edges; they where more or less the right shape and size to be modified to act as gun flints. One mans trash is another mans treasure.

The possibility certainly exists that washed up bottles from the west could have been salvaged prior to the arrival of colonists, but that's speculation unless a nicely sealed and dated pre-colonial context has glass in it that isn't locally produced.

Further reading:

Blume, C. L., 2010, “Making do with broken glass.” Mitsawokett to Bloomsbury Archæology and History of an Unrecognized Indigenous Community in Central Delaware, DelDOT Archaeology Series 154:245–264

Newland, M. D., & Meyer, M. D., 2003, Archaeological Excavations of the Old and new Russian Magazins Fort Ross State Historic Park Sonoma County, California. Final Report, Rohnert Park, California: Anthropological Studies Center, Sonoma State University Academic Foundation.

Silliman, S, 1997, “European origins and native destinations: Historical artifacts from the Native Alaskan Village and Fort Ross Beach Sites.”, IN: K. G. Lightfoot, A. M. Schiff, & T. A. Wake, 1997, The Archaeology and Ethnohistory of Fort Ross, California, Volume 2: The Native Alaskan Neighborhood, A Multiethnic Community at Colony Ross

Silliman, S, 2001,* “Agency, practical politics and the archaeology of culture contact.”* Journal of Social Archaeology, 1(2): 190–209.