How exactly would a situation like this work?

Lets say I have a website where the domain is registered in sweden. The actual hosting site is located in the Bahamas. The business LLC is registered in the bahamas. The business gets paid through paypal which is owned by the LLC and the LLC makes payments to an american.

The website has stuff on it that breaks some US laws. However, it does not break any Swedish laws.

A lawyer (from the USA) contacts the owner and says, "blah blah blah breaks xyz laws. Please remove it or we will sue you."

The owner responds saying, "sorry the site doesn't break any swedish laws. Have a nice day etc."

How will this situation play out? Will the lawyers just drop the case due to the hassle of prosecuting someone over seas?

Note the above is just an example situation I am actually curious how the law works in regards to foreign countries. I know the pirate bay has been sued numerous times yet their site is virtually impossible to take down. Same story for wikileaks.