The image is like a copy of the object (with appropriate scaling) at the location of the image. In particular, it produces the same depth cues as a real object would. If you can see both it and the lens with both eyes, simple binocular depth perception will tell you if it's in front or behind the lens. Another form of depth perception comes from motion (and works with only one eye): if you move your head a little, an image in front of the lens visually moves more than the lens, which moves more than an image behind it. Note that you can also get an image that forms behind the point where you view it, in which case it appears to be "further than infinitely far" and moves the "wrong way" in terms of parallax.
Also with one eye, and a bit different from the parallax based methods above: as you change the focus of the eye, the image will come and out of focus as if it were really at the location where the (extended) light rays meet, and not where the original is nor where the lens is. I.e. as you move your eye's focus from near to far, a real image projected in front of your lens will come in and out of focus before the lens itself, and the lens will come in and out of focus before a virtual image projected behind it.
The visual system instinctually understands all of these depth cue, though binocular vision is usually primary. That last point is an issue: lenses are usually too small to get good binocular parallax, so your visual intuition may be confounded. If you're good at controlling your eye's focus, I suspect that's the best way.
As a test, if you take a collimated beam of light and focus it with a lens, you should be able to judge the real image as a dot of light that hangs in midair in front of the lens.