Nosey people are invariably fiercely protective of their own privacy. If they had installed a nosey-parker conduit to get the scoop on your everyday conversations, surely they would realise that you'd also be able to hear them muttering and sharing their gossip - you might even steal their thunder if you overhear a particularly top-drawer newsflash about 'her at number 61 and her new fancy man' and run off to spill the beans before they can!
I very much doubt it was done for NP purposes. A jagged irregular hole would be much easier for them to deny knowledge of and blame on a mouse or something. The size and shape do suggest a removed or intended electrical installation. It could even have been a re-used piece of plasterboard that had previously been cut for a socket and used/intended for elsewhere.
Can you get a blanking plate put over it, or even just a tile or something? Or deploy a super-soaker through it (or even the aforementioned mouse) and see if they scream 
If they're such NPs, I'd be sorely tempted to make a recording with some juicy nuggets randomly thrown in and then prop it atop the cupboard, next to the hole before playing it loudly at regular intervals.
"I went to Tesco today as the queues outside Aldi were really long. Glad I did, really, as they had a really good offer on that wine we like. Wow, the hedge has grown massively overnight! I'll get the trimmers out and neaten it up later. Do you remember that time when Great Uncle Geoffrey criticised my baking and said my scones tasted like street cobbles? I'm so glad I had that old cast-iron skillet in my hand - he won't be doing that again! They said they wouldn't ask for a post-mortem as he was already 93, so assumed natural causes. Oh, the car will need petrol next time one of us goes out. It's been a little while since we checked the oil and tyres too, so that might be worth doing."