The chain of devices you need is: Microphone, pre-amp, processor, memory.
The microphone is easy. The CIA has been making odd sized microphones for years.
In a paper on low power microphones, researchers at Dartmouth designed a microphone pre-amp that consumes around 50uA. They did fabricate their device using 0.5um fabrication techniques, but they did not specify the size of the chip. Regardless, it should be quite small.
The 10 minutes of storage is tricky. If we use a simple processor that just shuffles the data from the pre-amp to memory, we'll need to store all of the data raw. The pre-amp above is specced for 4kHz sound, so we need 8kHz sampling. 8 bit audio will be fine, so that's 8000byte/sec, or 4.8Mb. On the other hand, if we ran a mp3 encoder, we could cram that data into 8Kbps mp3 streams, dropping our data needs to a mere 600kB.
We're going to need non-volatile memory, because the whole point is to store the last words. That means FLASH. The SST25 series is a low-power FLASH memory chip. It turns out that the energy usage of flash is almost independent of the size of the chip, so we can use the same device for both .wav and .mp3 versions... no need to spec them out separately. This chip uses 8uA when idle, and 30mA when writing. Fortunately, we don't need to be writing at all times. We can write in bursts to minimize power usage. For the .wav version, we'll be running at approximately a 10% duty cycle, so the actual energy consumption would be 3mA. For the .mp3 version, we would be running at 1.5% duty cycle, consuming around 0.5mA. These chips run at 3V, so they consume 9mW and 1.5mW respectively.
The processor is really the make-or-break component. A ENA2351 MP3 encoder proudly announces:
This product features a built-in
hardwired MP3 encoder/decoder system, enabling the industry’s lowest power consumption of 5mW...
The difference between powering a large FLASH full of .wav and powering both a FLASH full of .mp3 and a converter is pretty small. It's the difference between 6.5mW and 9mW. Because I think converting to .mp3 is a lot of trouble and the benefits are small, it makes sense to downselect here and decide to just store .wav data.
For processors, we can look at ultra-low power processors like a MSP430. This little 8MHz processor sports the Analog to Digital converter we'll need to get data from the microphone preamp, and also sips power, consuming just 0.4mA at full power (at 3V == 1.2mA), but it also supports several low power modes that may be used to decrease this.
In the end, we're looking at around 11mW of power (10.25mW rounded up to cover all of the unusual bits)
11mW is not much power, as far as the human body goes. The human body is capable of expending 20,000 times that much. However, getting 11mW parasitically is tricky. So far tools like TEGs, worn on the outside can generate 0.02mW/cm^2. Enzymatic Fuel Cells inside the body are similarly on the microwatt range for power output.
You do have access to a lot of unusual power sources, of course. If you're implanting a microphone on a person, it makes some sense to couple to the diaphragm. You may be able to get your 11mW by hooking to the diaphragm in some manner and leveraging their breathing. They're guaranteed to be breathing all the way up to their death (or shortly before their death), so it's a very reliable source of power.
Other than this incompletely specified power source, everything I've described could be fabricated on one or two chips. Everything could go on one chip, system on a chip style, or you might keep the flash memory separate because it can be mass produced better that way.
As for what it would sound like, that's open. It depends heavily on where the microphone is. If you put it in the chest, you'll find that it sounds like a doctor's stethoscope listening to your breathing. You would probably want the microphone closer to the mouth to capture the phonemes as they are created. It most likely would not catch anyone else's voice because the dynamic range would be too great. Remember, your own voice is so loud that your brain unconsciously moves the bones of your ear further apart just before speaking. That's done so the sound of your own voice doesn't make you deaf.