Customs normally does destroy illegal or copyright infringed goods if it finds them and can identify them as illegal. But customs may not check all shipments and may not find all illegal items and just because something "gets through" customs provides no protection from prosecution from selling illegally copyrighted items without the consent of the copyright owner. Normally any item which would be illegal under copyright law to make in the US is illegal to import for sale into the US (see here). In addition, most agreements with companies that allow others to sell goods through them include statements saying that the seller agrees he/she is only selling legal items (meaning those that do not violate any copyright or trademark or patent laws or others) and, if it later turn out that he/she is not he/she will be responsible for any attorney costs of defending the company and any legal damages the company may incur due to fines or other legal costs. That normally means, again, that just because a website allows goods to be sold on their site that also provides no "proof" that what is sold must be legal.
The seller of goods that infringe federal copyright laws are liable for a federal crime as copyright is a federal law and may be forced to pay fines and even for potential imprisonment. In addition, they may also be required to pay the copyright owner any profit or income they have earned from the sale of such goods.
Normally, when U.S. retailers buy products they are responsible for assuring those sellers are licensed or legally able to provide those items to them. If they make any mistake they will normally suffer the consequences. Traditionally, under U.S. civil and criminal laws all parties who manufacture, wholesell, and retail sell an item are responsible to the consumer if the product is dangerous or illegal (for copyright or trademark reasons, for example) and the consumer and/or the legal authorities can go after all of them.
If a shipment ito a retailers is from a U.S. company, known in the industry, then normally those items are known to be legally sold or available by them and from them based on reputation and their tradition of having exclusive retail agreements or limited resell agreements with the manufacturer or distributor. Smaller retailers themselves may not always investigate whether everything they buy to sell is from licensed distributors or wholesalers or sellers, but that does not mean that they will not still be held liable if anything they buy and then sell turns out to be illegally distributed.
Knowing how to verify the authenticity of a product will not always be done the same way. Much will depend on the item, the industry, and traditional practices in that industry. For example, in the movie business the Motion Picture Association of America may be able to provide information about how to recognize properly copyrighted material and whom they license to sell it as that business tends to be very controlled about whom they sell or allow to manufacture their items. But, each industry is different so its a matter of being able to identify the particular factors in that industry that easily label something as copyright legitimate or not. In other industries, such as the art world, it is a matter that only certain experts may be able to identify.
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