Only if it's poorly designed (leaks local oscillator frequency back into antenna) and not shielded.
Modern receivers use quadrature sampling detectors rather than traditional superheterodyne. In that setup any leakage would be on the same frequency and harder to detect.
It is as good as impossible to run an oscillator based receiver that is also connected to an antenna that does not radiate. That's nothing to do with poor design, it's just physics. Zero coupling does not exist in practice. By design the mixer stage sits pretty close to the initial amplifier and it will result in some of the oscillator energy making it back to the antenna circuitry. FWIW I built a ton of transmitters and radio gear in my teens, it is pretty easy to take a theoretical stance here and declare that anything that leaks is not designed properly but that's about as 'true Scotsman' as you could get.
Note also with a highly directional high gain receiver that tiny bit of radiating energy is very detectable. It's just going to feed into background noise for most receivers so no one cares. But it will be detectable by a motivated hunter with the right equipment.
A directional antenna has to be much larger than wavelength. Won't fit in a van if we're talking about HF (with wavelengths measured in tens of meters).
Not[0] really[1]. Loop antennas (active or passive) have good gain perpendicular to the loop. It would be easy to mount on the inside of a wood paneled moving van with some nice receiving equipment. You park the van so the side faces the target's apartment and voilà you can listen to the mixer of their superhet radio. If your searching for receivers a couple vans driving around could triangulate and detected signals. Just getting the right apartment building would be enough for the KGB to turn the place over.
If you've never built a radio and tried to shield this unintended export than I can totally see how you might think this is just a matter of careful design and more shielding but it really is a lot harder than that and you will simply never reach zero to the point where even an ordinary spectrum analyzer hooked up to the input of your radio will not show the oscillator frequency as a nice fat peak.
There is a big difference between spectrum analyzer hooked up directly to the input of the radio and an antenna in a van several blocks away, with reflections and noise thrown in.