Well, If you are importing from a wholesaler or manufacturer, the responsibility for Customs documentation and import duties is one of the things you agree upon when negotiating the shipping terms. The negotiation part is actually important since this will determine if you can agree on how they will facilitate you on importing your products, or perhaps they can help in directly shipping it to your customers.
Most of the time it will be your responsibility to organize Customs clearance and the payment of import duties.
This will involve hiring a freight forwarder or customs broker, both of whom will work out how much duty you need to pay and will make sure all the appropriate documentation is filed. You can find lists of freight forwarders/customs brokers in your area by doing a Google search.
e.g. "[california] + freight forwarder".
For smaller imports you may prefer to use a shipping company such as DHL or UPS. These services will include duties in the overall shipping price.
If you are drop shipping to an international customer then your customer is responsible for paying any import duties. It is very important that you note this in your auction listing so that the customer is not taken by surprise!
It is also a good idea to check out the Customs rules and regulations for the countries you are going to ship to in advance. Make sure that they actually allow the product to be imported and that duties aren't going to be too expensive.
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I hope this helps. :)