In the USA, does fish being served raw need to be frozen first?

Yes, it is true. Fish other than tuna must be frozen in the US to be called sushi grade. Only sushi grade fish can be sold raw in restaurants in most (if not all) jurisdictions. Freezing kills parasites that are common in fish. The FDA recommends that fish for raw consumption be frozen first, but state and local jurisdictions make and enforce the law. My understanding is that even in rare jurisdictions that don't have the law per se, no one risks the liability inherent in violating the recommendation.

The only concern any inspectors have is referred to as the parasite destruction guarantee, which is accomplished by ‘freezing and storing seafood at -4°F (-20°C) or below for 7 days (total time), or freezing at -31°F (-35°C) or below until solid and storing at -31°F (-35°C) or below for 15 hours, or freezing at -31°F (-35°C) or below until solid and storing at -4°F (-20°C) or below for 24 hours’ which is sufficient to kill parasites.

Sushi FAQ

I'm not sure if Japan has any similar regulations, but I do know that fresh raw fish is frequently eaten in Japan, often in homes, and that Japan has a much bigger problem than the rest of the world with certain parasitic infections of their people.

Anisakiasis has recently become a leading problem in Japan. Anisakiasis is a disease caused by an infection of Anisakinae larvae, which are the visceral larva migrants following an infection by Anisakis larvae or related nematodes. It occurs when the live larvae are taken into the human gastrointestinal tract by eating raw, infected fish, in which the larva is in either stage III or IV. It causes pain, and often the pain is so severe that the patient is treated for acute abdominal pain, although the disease is by no means fatal.

General Survey of Anisakis and Anisakiasis in Japan

If you follow that link you will see a map of Japan and the numbers of cases of anisakiasis reported by hospitals. It doesn't give a time frame for the reports in the free portion of the study, unfortunately.

Japan alone accounts for 90% of all cases of anisakiasis described in the literature because of the widespread use of raw fish in traditional Japanese cuisine, with sushi and sashimi.

Abstract in the International Journal of Infectious Diseases

If you are a visual learner, you might be convinced to only eat raw fish after that fish has been frozen per the above FDA minimums by looking at this abstract with pictures from the Korean Journal of Parasitology.