Hi Gilligan and the castaways,
So far Ginger doesn't know, Mary Ann is just too cute for cognative thought, the Howells only know about money and the skipper is, well, the skipper. That leaves you with the professor but can Little Buddy fathom the answer?
Put de lime in da coconut an' drink 'em bo' to-gedda.
Uh, sorry, wrong island. Now let's get down with the local oscillator that heterodynes with the incoming signal in the mixer stage following the RF amplifier to produce the first intermediate frequency (IF). That's the theory they're stabbing at but don't quite have it right, that oscillator may radiate a weak signal for a very short distance, a couple of feet at most. In a worst case scenario (really crappy receiver without an RF amplifier stage) it may back up through the antenna and go a bit farther but not by much. This is what causes some scanners sitting side by side to lock up on a "dead carrier" that some incorrectly call a "birdy" which is actually a spurious product of a crappy receiver, it locks up on itself.
Now you have the "theory of operation" let's take a look at the radio you're trying to figure out. Like I said a scanner for example might radiate for a few feet, a proper receiver, ham, commercial, etc. won't radiate at all. There goes your stupid scanner trick, myth busted.
Someone mentioned radar detector detectors but first let's examine the front end of a receiver. In a good receiver the signal passes through an RF amplifier before entering the mixer where the local oscillator signal is injected. The mixer output passes through a tuned circuit which selects one of four mixer products and sends it ti the intermediate frequency amplifier. In a dual conversion or triple conversion receiver this process is repeated two or three times each producing a lower IF and thus greater image rejection as the succeeeding tuned circuits filter them out.
Please note the RF amplifier stage effectively isolates the mixer and local oscillator from the antenna so far less if any local signal is radiated than when no such stage is used. This was the case with the famous "All American Five" 5 tube AM table radios of years gone by when a combination oscillator and mixer called a converter was fed by the antenna directly. Cheap "transistor radios" used the same IF conversion scheme as did many cheap shortwave and VHF receivers which were notorious for television interference, they radiated like little transmitters. I know, one was my first "scanner" back when they had tuning knobs and radio dials.
Now we FINALLY get to the radar detector detectors, whew. The crappy old radar detectors operate on the same principle using a Gunn diode self-oscillating converter mounted on a panel antenna which radiates a strong signal. This signal is picked up on a sensitive microwave receiver and it's relationship to the radar frequency reveals it as coming from a radar detector rather than some other microwave source.
Just an intersting footnote, early Neilsen TV ratings polls were done electronically from a truck going slowly down the street. It contained a sensitive receiver and spectrum analyzer hooked up to a recorder and used the signals from the local oscillators in TV tuners to log which stations they were tuned to and whether or not they were operating. At the time nearly every home had a TV set so with statistical error correction a reasonably accurate survey could be taken.
That's your lesson for today dear castaways, now get busy with the vines and coconuts and build a generator to power the transmitter I just made from Ginger's transistor radio. While you're doing that I'll be wondering why the boat engine never had an alternator and battery system or that fat pork never put in a marine radio. What a stupid show, at least Ed Wood's movies made sense, sort of.