There are different kinds of night vision devices, and the answer to this question depends entirely on which kind you are talking about. Each kind is based on a different method of "seeing" in the dark, of which you've given only one.
For that kind of night-vision device - which works by illuminating a scene with light that is invisible to your eyes but visible to a camera and which thus incorporates one or a pair of such cameras together with a suitable light source, then yes: if the two goggles are identical, they will be able to see each other's "lights" as "headlights" in the distance, because both emit the same kind of light and the detectability of a certain kind of light does not depend on its source or origin, only on its physical characteristics as an electromagnetic wave which means its wavelength and intensity (brightness).
However, this particular kind of device is rather limited: for one, such a device's "night seeing" power can only go as far as its beam can, meaning it can only let you see things proximately. It won't, say, alert you to a bear that's 200 m away from you. It's effectively a flashlight that you can hide from people who don't have one (and from most animals, as well, at least I think, since the infrared in the wavelengths employed for this purpose are not wavelengths that typically occur enough naturally in ample enough quantity to be a useful thing to evolve vision for). For another, as you've noticed, it effectively will "give away" people who are using it to other people who are using it, which would make it problematic in, say, a military situation - though it still could be useful for many non-military applications.
The reason these exist is that they're cheap: detection of infrared wavelengths between 750-1100 nm, approximately, is easily done using a conventional silicon-based CCD sensor like that in your digital camera. Such sensors, due to mass production, are cheap and readily available, and thus the goggles themselves can be made for cheap this way as well.
The other two kinds of night-vision devices (and there may be more) are those based on passive imaging: they are devices which do not use a light source of any type, but instead rely on sensing pre-existing radiation that just happens to be outside the range of our eyes to see, whether in wavelength and/or intensity, yet is still present when visible radiation is not. Naturally, because these require specialized sensors made for this purpose, they are considerably more expensive than the first kind.
One such sensor is based on the principle of looking at waves of much lower amplitude (intensity) than our eyes can see: it is what is called an image intensifier. It is really just a very, very sensitive light sensor that is capable of reliably picking up extremely low levels of visible wavelength light - too low for our eyes, but still present during most night-time conditions. These are the ones that give the "green" images most often associated with night vision and seen in computer games, which is then hooked to some sort of display screen. In this case, yes, you will be able perhaps to see the other person and vice versa, but not because the fact of the other having goggles makes them more visible to the one, but rather because that's the whole point: to let you see things at night, and that would include people, regardless of whether they do or do not have goggles.
The other kind is also an infrared sensor, thus based on the "use wavelengths outside of human vision" principle, but using a much deeper wavelength than the kind of goggles you are talking about, that is in the mid-wave to long-wave infrared (MWIR / LWIR) areas of the spectrum, about 3-5 μm and 8-12 μm, respectively (in nm, that's 3000-5000 and 8000-12 000 nm, so you can see these are MUCH bigger waves than visible light waves which end at 750 nm at the longest.). These wavelengths are long enough that objects at around room temperature emit them due to thermal radiation - it is just like how your hot stove burner glows red visibly, but cool objects do just the same except in wavelengths far too "red" for your eyes to see, yet these cameras can do it. Since human bodies are generally warmer than their surroundings, they will be easily visible on such a camera. Visibility concerns with regard to other users, then, operate in exactly the same way as for the ultra low-light goggles/cameras.
TL;DR, potentially yes in all cases, but the presence of goggles on the other person is only a modifying factor with devices like those you mention which actively shine some kind of invisible light from them.