Do illusions exist as images and sounds in the world, like holograms, or do they exist only in the beholder's mind, like mass hallucinations? There are paragraphs in the PHB that can support both assumptions.
Clarification: the question primarily concerns illusions produced by Minor Illusion, Programmed Illusion, Silent Image, Major Image, Disguise Self spells. Spells like Fear or Phantasmal Killer explicitly says they affect creatures, so they are out of the scope.
Illusions as actual images and sounds
Spell description explicitly says that the caster "creates a sound or an image":
You create a sound or an image of an object within range
When casting an illusion, the caster have to specify a location, not a creature. Anyone who looks at the location perceives the illusion.
Many obviously mind-affecting spells like "friends" or "charm person" are enchantments, not illusions.
While there are creatures immune to charm, there is no creatures immune to illusions. Even a construct can perceive an illusion.
Illusions as mind tricks
The wizard's arcane tradition describes the School of Illusion as
magic that dazzles the senses, befuddles the mind
Being revealed, an illusion looks different to the perceiving person only:
If a creature discerns the illusion for what it is, the illusion becomes faint to the creature.
When does it matter
There are several cases when the outcome might depend on where the spell effect is located:
Antimagic Field – to be negated, should the illusion "itself" be in the field, or does any creature in the field become immune to illusions?
Detect Magic – assuming that Detect Magic spell can detect illusions, what exactly should be in the 30 feet range to be detected?
Also there are ambiguity with light, vision and line of sight:
Block light – A caster tries to use an illusion to cover a window, or conceal a lit torch.
Block vision – Many cases. A caster tries to hide an object "under" an illusion, but the spell description says it can only "create an image" and not "hide an object". If we assume an illusion actually can cover an object or a creature, Disguise Self would protect from all LoS-dependent spells, etc.
Best Answer
They are "real images"
Illusions can create either a "real image" or sensory experience that anyone can perceive, or plant images directly in the mind of creatures.
This implies that "phantom images" created such that any creatures can see them (Silent Image, Programmed Illusion), are distinct from illusions that plant images directly in the mind of a creature (Fear, Phantasmal Force). This further implies that the images created by silent image are not planted directly in the mind of its observers.
Mechanical Perspective
The spells minor illusion, silent image, programmed illusion, and disguise self all have a range.
Let's take for example, the spell disguise self. It has a range of Self and can change the appearance of the caster superficially. These magical alterations, though limited only to the caster (as seen in the range of the spell), affects the perspective of anyone who can see it -- even from hundreds of feet away, if visibility allows.
Similarly, the spell silent image can create the image of an object within 60 feet of the caster, affecting an area no larger than a 15-foot cube by creating a visual phenomenon in that space. Therefore, the spell does not affect the minds of its observers (at least, not directly), but rather only a specific spot within range of the caster.
So, if someone 300 feet away looks at this image, they should still be able to see it, since they aren't being directly influenced by the spell. They're just perceiving a space differently, and that space is what this spell is affecting.
The same arguments can be used for programmed illusion and minor illusion. These spells create illusions that exist outside the mind of their observers.
The "paradox" of different people seeing different things
Once a visual illusion is discerned, it becomes faint to that creature and to that creature only. Meanwhile, others continue to see it as if it was really there. This imparts to these illusions properties of a mass hallucination, because two people can look at the same place and see different images (one sees a "full" image, the other sees a transparent one).
Strictly speaking, from a mechanical PoV, these images are external to the observers. So there should be an in-world explanation for why they have properties that suggest they are hallucinations instead.
This can be explained any number of ways, though, without being cornered into accepting the illusions as directly affecting the mind of the observers. First of all, it's magic, which means it doesn't have to be intuitive. One way to explain it is, since these illusions affect a space (not you directly), and you are just perceiving that spot differently, the Rules of Magic could change depending on your knowledge as the observer. There are other spells which have this property too: zone of truth does not allow you to deliberately lie, but you can say something that is false if you believe it to be true; geas could allow you to act against the direct instructions of the caster if you don't know you are violating those instructions (otherwise you could use geas to say "Say only things that are true about X" and everything the target says will have to be factually true about X, even if they had no way of knowing if it was true otherwise).
Another way, which is simpler, is to say the spell could be presenting you with an optical illusion, one whose nature is such that once you see it, it's hard to "unsee."
Implications
Antimagic Field
The spell says:
This clearly means illusions that create images in the minds of its targets are absolutely nullified if the target enters an area affected by antimagic field. But they can still see an image (created by, say, silent image) that is not in that area, for ex., one just standing outside the border, because that image is not inside the area affected by antimagic field.
Detect Magic
The spell says:
If a magical illusion is within 30 ft of the caster, then the caster will detect it (but they won't necessarily know where it is, or which school it's from). If a phantom image that anyone can see enters the range of 30 ft, then they will sense magic is present. If someone affected by the fear spell enters their range, they will also detect the presence of magic.
Light
This is a messy topic, and is best left to DM adjudication.
If an illusion cannot block light, then it must be invisible -- you would see through it because, to see in the first place, you need to see the light coming off it. If it can't block light, then that means you are still seeing the light coming from the object directly behind it, and the illusion is covering nothing.
On the other hand, if an illusion can block light, then illusion spells which say: "[p]hysical interaction with the image reveals it to be an illusion, because things can pass through it" are contradicted, because light is a thing. If light, as a thing, can't pass through the illusion, then the clause on illusions being unable to block things is violated.
So the implications for light, as far as illusions go regarding "are they visible to all, or only in your head?" are messier and tangentially related at best. It is probably applicable only to spells which conjure phantom images that anyone can see, though, and not to illusions that plant images directly in your mind.
Vision
Line of sight is defined in the DMG:
If something blocks your vision, you do not have line of sight. If an illusion obstructs your vision -- ie, you cannot see what is behind that illusion -- then your line of sight is cut.
Notably, even a thick curtain can block line of sight. Thick curtains probably cannot shield you from an arrow or a beam created by disintegrate, but it can obstruct the vision of those who would use these things against you, and that is enough to stop them from targeting you.