So, what I wanted to post has mostly been covered, but I wanted to get this down anyways, as a sort of collection of all the different ways of achieving invisibility, their effectiveness, and the invisible person's ability to see.
There are five main methods to achieve invisibility:
Camouflage Invisibility
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The easiest method, the one that we're most familiar with, and the one with the most real-world examples. Most people wouldn't consider this "real" invisibility, but it is nonetheless a very effective means of remaining unseen. Animals have been employing this method for milennia to keep themselves alive, so there must be something to it, right?
Basically, this method boils down to having your physical appearance mimic that of your surroundings. Whether that's from wearing clothes designed to blend in, or by changing your skin color and patterns to match.
The Downsides: Compared to the other forms of invisibility, this one has many downsides. Firstly, as mentioned above, you're not actually invisible, just very well disguised. You still cast a shadow, your shape can be distinguished, and your disguise is only effective from certian angles. You also need to either keep still, or move very slowly. Your disguise is also limited in that it provides only a general blending - it cannot display exactly what is behind you. If your backdrop is very specific, or changes rapidly, you'll be much easier to spot. Also, if you're using clothes or other static apperati to become "invisible," your options for where your technique is effective are limited.
If you're using an adaptive camouflage, whether biological, mechanical, or supernatural in nature, would allow you to move quicker and have a greater range of backdrops to hide behind.
Can You See?: Yes, you can! Light is not prevented from reaching your eyes, and so your vision will be unhindered. however, in most cases, your eyes will not match the environment you're trying to hide in, and so might break the illusion and give you away.
Transparent Invisibility
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The one that, I believe, most people assume when considering invisibility, transparent invisibility means that light simply is not disturbed by your presence, and so you cannot be seen.
There are examples of transparent invisibility in the real world (or near enough), only most of them do not involve living things. Glass, water, and other substances are nearly transparent. They do, however, still refract light, which creates a distortion effect that is noticeable.
The degree of transparency determines how invisible you are: a low transparency means you would appear as see-through, but still be visible. Your transparency would be different than how much you refract light, however, and you would still be detectable (although difficult to pin down) by the distortions you create, if you refract the light passing through you. "Perfect" invisibility with this method requires 100% transparency and no refraction.
The Downsides: The main downside comes with the aforementioned refraction. If you refract light, the distortions would give away your position. Additionally, since we get a fair amount of our external heat from light striking our bodies, being transparent would get very cold very quickly!
Can you See?: This depends. If your transparency is less than 100%, then yes, you can see. However, your vision would be restricted, almost as if it were much darker than normal, as only a fraction of the light you would normally see would reach your eyes. With perfect transparency, even if you refract light, you would not be able to see.
Distortion Invisibility
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Also known as the "Invisible Girl" method, this type of invisibility involves bending the light around the invisible subject, and then bending them back into position after they've passed, creating a pocket around the subject. There are very few examples of this in nature, but we've been able to bend light for a while, so a device that could create this effect could be theoretically possible, if not necessarily practical.
Unlike transparency, distortion invisibility doesn't involve to what degree you're invisible. Instead, it requires you to think about how large an area whatever is creating the distortion needs in order to bend the light around you effectively. the larger the area, the less power would be needed, but the further from you a potential observer can be before they notice something is wrong. If the field is not big enough, you would become invisible at the edges, and more and more visible (in a hazy and distorted fashion) the closer to the center of your image, from the ponit of view of the observer.
The Downsides: The major downside of distortion invisibility is that the distortion field must exist outside of you. Therefore, once a potential observer approaches to near the field, they would start to notice that something is amiss. When they proceed within the field (if they can), they would become effecitvely invisible as well (and if, they had any light sources, would be able to see you).
Additionally, the same considerations to heat as in transparent invisibility must be taken into consideration.
Can you see?: Yes and no. The distortion moves outside light around you. None reaches your eyes, so you would not be able to see outside the distortion field. Light sources inside the field would allow you to see inside the field. However, this could be risky - depending on how the field works, that light might be able to escape, giving away your presence.
Projection Invisibility
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Projection invisibility is how most of our current "invisibility" attempts work. In effect, the light that strikes the object to be made invisible is recorded, and then a matching beam of light is projected from the other side of the object, creating the illusion that the object doesn't exist.
There are no examples of this in nature that I am aware of.
The Downsides: For a device that prodices this effect, to achieve the best invisibility possible, there would need to be a matrix of sensors and projectors covering the surface of the entire object, which is technologically challenging. Additionally, because the sensors would not be able to project, and the projectors would not be able to sense, the projected images would be low-quality, potentially giving away the presence of the object. A supernatural verison of this effect, however, would have no such limitation, and could theoretically proceed upwards to perfect invisibility.
Can You See?: Yes! At least, probably. A large object, such as a car or other vehicle that you could be inside could be rigged to project the images gathered both inside as well as out the opposite side, allowing you to see the same image that observers on the other side could. An outfit that comes equipped with a helmet or visor could also project a similar image. With a supernatual effect, the light is still striking your eyes (and being re-projected out the back of your head), so you would be able to see normally.
Phantasmal Invisibility
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The only version in which light is not actually being manipulated, phantasmal invisibility is entirely supernatural. You use a psychic illusion to convince observers that you aren't there.
There are two main versions of this kind of invisibility: in the first, you convince the observer that you are uninteresting and boring, or else supposed to be there and innocuous, and so they ignore you. The second has you trick the observer into not seeing you at all.
In the first version, you could even manipulate objects in plain sight of observers without alerting them, as long as you do not draw undue attention to yourself. In the second, such things would likely alert them, although if you can trick their mind to think that you aren't there, you can probably also trick them into thinking the objects haven't moved.
The additional benefit of this method is that you could extend it to other senses beyond sight. You could become inaudible, unsmellable, and potentially intangible, as well as invisible, because the entire effect is in the minds of the observers.
The Downsides: In either case, you would still be observable by inanimate observers, such as cameras. If you are using the ability to make yourself boring and uninteresting, performing actions that break that illusion could result in the effect ending. With psychically-induced invisibility, such things would not break the effect, but still might alert observers that something is amiss.
Can You See?: Yes! No light is being manipulated, so you can see perfectly normally.