This research is fraught with technical difficulties. I agree that it is likely that a shielded enclosure (Faraday cage) will be required, and I think it likely that it would have to have an anechoic interior. My work in testing undesired EM emissions showed me that reflective internal surfaces in a shielded room or box destroys the accuracy of your measurement, and that anechoic or open space is required. Finding a suitable test site is challenging.
I also wonder about whether electric currents in the body that originate biologically would indeed modulate RF current coupled into the body from outside. Normally, we must have a non-linearity of some sort for modulation to occur. Without such non-linearity in the medium, the biological electrical activity would not mix with currents to create new spectrum, and this is essentially what modulation is, the creation of new spectrum around a carrier (for example). This begs the question of whether there is any human tissue that behaves non-linearly in its reactance to small RF currents. I have some doubts about this, although, with many media, you can get non-linear effects when the field strengths (induced currents) imposed on the media are high enough. This suggests that the previous experimenter was using some pretty strong RF on his test subject. You noted that someone believes they have already detected re-radiation of modulated RF. Do you have a copy of the publication for review?
In your previous experiments, you were using a diode detector. These are notoriously insensitive compared to a good radio receiver, so if you did indeed measure radiation from the body, then the field levels must have been fairly high. But others don't find such high field strengths as you do, so I wonder whether your detector was, in fact, responding to the ability of the body to affect the field strength of electro-magnetic fields that originate elsewhere. I am interested to know what kind of antenna you used for those initial tests with your detector. Can you provide a photo or description? I think the effect you see is most likely that your body is distorting the EM field around your detector antenna when it is close, simply by virtue of having a very different dielectric constant and conductivity than air.
It is worth mentioning that many spectrum analyzers have relatively poor noise figure and require external pre-amps for sensitive work.
It is commonly done to bias diodes with a bit of DC to improve their sensitivity as a detector, so you are perhaps on the right track thinking that house AC may be helping you. On the other hand, by adding an amplifier in front of your diodes you may be creating new trouble unless there is some sort of band limiting filters to help exclude undesired EM from outside sources. Adding an amplifier to a diode detector is the first step in creating a radio receiver, and I wonder whether you want to re-trace the steps of many radio pioneers, as this would be a long road to follow.