Ha! I have an answer. (these steps only work for the P1000)
It took many questions, with no real answers, and a LOT of experimentation.
This is completely unintuative if you're a beginner, but relatively painless - follow these steps:
1. Unplug the HDMI cable. (You need to do this otherwise the next step won't work)
2. Menu > Spanner (setup) > HDMI > Clean HDMI
You need to turn this option ON.
3. Exit the menu.
4. Plug in your hdmi cable, and connect to a suitable monitor, recorder or whatever.
5. The internal camera display should go blank, and the external monitor (or recorder or whatever) will now show what the internal monitor would have shown.
6. Press the Display [DISP] button repeatedly till you get a clean display.