I have a website where people need to be able to clearly see each other. What is the smallest that say, a PNG, could be, while allowing the viewer to clearly see the features of a face?
To get this out of the way up front: "image size" could mean the physical size at which the image appears on the viewer's monitor, or the dimensions (number of pixels wide and tall) of the digital image stored on the computer. Since the question asks about a PNG file, I will assume the latter of the two, but understand that they both play a role.
My subjective answer to this question agrees with what dpollitt says in his answer. The common internet avatar sizes are 80x80 px. and 100x100 px., with some others like 48x48 and 73x73 (twitter), 96x96 (Gmail), 158x158 (Myspace), etc (jobmob.co.il); there are also some non-square sizes such as 48x64.
For a more scientific answer Photo.SE may not be your best source, but you could always search for academic studies on human perception of low resolution images. I'm sure there are a number of them out there.
I would stick with the Gravatar standard size of 80x80px. It is pretty common for people to use a headshot for this, and faces are easily recognizable. I don't know the motivation for why they chose 80x80 as the size, but I would assume some thought went into it, and they realized that people many times use a headshot and want to be recognized.
I would give people the opportunity to change the size, because it will vary based on the resolution and screen size. And then make sure that the image quality is reasonably high - 200x200 has been suggested, but you might have to experiment a little.
The avatars for SO are too small at 32x32. However, I would probabl yjsut about recognise a face on one. It is remarkable how good we are at recognising faces that we know. The real issue is whterh they have to recognise the faces, or distinguish between faces. The requirement is different in each case.