[klondike98]

Here's a thread on the AM Forum about the topic.  It mentions a proclamation that all citizens, natives etc. of a hostile foreign nation could not own short wave radios (among a number of other items like weapons, bombs, signal devices etc.).  

On the Museum of American Heritage web site there is the following info on WWII Radio servicing that includes a comment on German and Italian nationals being forbidden from owning a short wave radio.

"Radio Servicing During WW II
In 1942 the manufacture of radio sets for civilian use was stopped and the facilities were directed to war production. By 1943 sets were in short supply. Old radios, long considered obsolete, were resurrected from basements and attics and put into use. The owners needed to bring many sets to the local repair shop for restoration It was quite common to find the remains of a dead rat or mouse that had crawled into the chassis to get warm only to be electrocuted when the set was turned on.
Repairing these sets was at times a dangerous occupation. Capacitors contained PCB's; cabinets had asbestos insulation; the solder contained lead; wire insulation was often poor -- and on top of this, the internal operating voltages often exceeded 350 volts!
While most radio component production went to the military, the government did allow some components to enter the civilian market. These were marked "MR" for "Maintenance and Repair." During these times it seemed that there was a radio repair shop on every block of a large city. Many of them had poor reputations, for some repairmen were out-and-out crooks, often rewiring sets for substitute components at exorbitant prices when originals were readily available.
With tight gas rationing (an "A" sticker allowed only two gallons of gasoline per week), many car owners removed their car radios and had them converted into home receivers. German and Italian foreign nationals were forbidden to own radios with short wave bands for fear that they could be used for espionage purposes. They were required to bring their sets into repair shops to have these bands disconnected.(1) [emphasis is mine]  
(1)  www.moah.org/radioking/radiokiosk/early3e.html   and www.moah.org/radioking/radiokiosk/early3f.html