It's weird to think that we need a mirror (or other means of imaging) to see our own face. We can see a large part of our own body directly, but we will never be able to directly see our own face like other people do (unless, I guess, you have an eye pop out in some kind of accident, but I guess at that point this is not the kind of thing you would consider).
You also can't hear your own voice like other people do, because what you hear when you are speaking mostly come from the vibrations in your bones rather than from sound waves going through the air.
And unlike with mirrors, something we always had, be it just a pond of water, the ability to hear your own voice like others hear it came with recording devices, and is therefore very recent (19th century).
You can hear your own voice as an echo, though it is somewhat distorted in a similar way as one's reflection is warped in a pool of water. Amphitheaters are an ancient building technique which utilize acoustic properties to reflect sound toward distant listeners, but I'm not familiar with structures built to reflect sound back toward the speaker specifically, though I imagine that such a structure could be built. Naturally occurring echo chambers exist in the environment, as well.
