I've figured out what's going on with the audio.
Summary: One channel of the story portion of the audio is inverted. Listeners having problems hearing it are listening to it on players that don't support stereo and are combining the left and right channels into almost nothing.
The whole story (Only of interest to the technically-minded):
I was curious about the problems, so I played the episode on my usual player*. It played both the story and the other parts fine. Then I played the track on my computer (with the VLC sound player). No problem yet.
Then, while still playing the file on VLC, I pulled up the mixer software for my sound card**. I played with the panning controls and found that, during the story portion of the cast, if I panned either channel to the other side, (i.e. if I panned the left channel to the right, or vice versa), I got the effect that others have commented on. The effect didn't occur during the intro/outro (not just my voice, either - also the background music, Peter's voice at the very beginning and Ann's bit at the end).***
I opened up the file in Audacity, to see if it looked weird at all, but could see nothing obvious. Since Audacity has to convert the file from MP3 format to raw audio samples, I suspected that the problem lay somewhere in the MP3 encoding, but when I tried converting the file into a WAV and then back to MP3, the problem didn't go away.
So, in an attempt to see what would happen if I only listened to one side, I decided to try splitting the stereo track into two mono tracks. However, I made a mistake and instead, combined the two tracks into a single mono track.
It was a fortuitous mistake. Jackpot! The result was a track with exactly the problem characteristics.
I undid my error, and took a closer look at the audio from the story reading and realized that the contours of the two channels were almost an exact mirror of each other, rather than (as is usual) being similar in shape. So when they were combined, they (nearly) cancelled each other out.
I think what's happening is that the people who are experiencing problems have players that don't support stereo. When the player gets a stereo track, it simply combines the two sides into mono, and if, as in this case, the contours of the two channels mirror each other, they cancel out (or nearly so, since the mirroring isn't exact).
If Peter is willing/has time, the solution is to
a) split Ms. Croft's original stereo recording into two separate tracks,
b) invert the audio of one of the tracks (Audacity can do it, so Peter's software probably can),
c) combine the tracks back into a stereo track.
(And then, of course, re-edit the episode back together again and re'cast it. Unfortunately, if he did this procedure on the whole episode, the non-story parts would suffer from the same problem...)
I have to say that I'm curious as to how the track got this way. I'd be interested to know what software Ms. Croft used when she recorded the story (especially whether it's a new program, or an upgraded version of an old program, that she's started using since she recorded "No Better Than a Beast", as I don't think anyone reported any problems with that).
* My MP3 player is a cheap, generic one, with no support for the DRM in the likes of iTunes, Audible, etc.
** My sound card is an M-Audio 2496.
*** I have no idea why the sound card's software allows me to do this. But it does.