The cloaking device in Star Trek is imperfect.
If you look closely either with eyes or sensors you can observe a disruption where the ship is.
This is because it isn't successfully bending all of the light around the ship.
Some of it is going into the ships sensors, being absorbed or being reflected at random.
And this creates a detectable disruption, but it also means that the ship can still see.
It's only with total invisibility that all of the light has to go through and you'd have issues with seeing.
This might apply to the invisibility question as well.
The only light that has to go through you to be invisible is the light coming from behind (wrt the observer).
As long as you don't reflect the light coming toward your front (wrt the observer), then you won't be visible.
Whoever is looking at you will see only the light coming from behind you.
Of course, this is an issue if there are people on all sides of you.
However, if you only allow certain points to absorb the light instead of bending or transmitting it, then for all practical intents and purposes, you'd still be invisible.
There'd only be a very, very slight disruption right around the sensors.
Or eyes.
The bit about not reflecting on coming light only works in space actually.
Unless you count 'casting a shadow' as being invisible.
Which sometimes happens and sometimes doesn't.
Depending on the story.
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I'd like to propose another answer.
Looking at the cloaking technology itself, there are two ways that it could work: First Case:
Light hitting the ship is absorbed and emitted on the other side of the ship without alteration from all directions.
In this case, your technology is actually capturing the light, so it IS "seeing" everything as normal...
it's just not reflecting it back for everyone else to see them.
Problem Solved.
Second Case:
Light hitting the ship is bent around the ship and then straightened again on the other side without any other alteration by the ship.
In this case, you're going to need a small inconsequential amount of light to pass through the cloak barrier and be absorbed (not reflected) by the ship as sensor data.
So the bottom line, as long as you have technology that absorbs light without reflecting it back...
you can "see".
But the more light you absorb without passing it to the other side, the more distortion there's going to be... and
that is a way for the other ships to detect you.
(Of course, any energy being emitted by the ship is detectable...
so you have to turn the ship into a closed sealed system).
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Out-of-universe, it's easy to explain: The writers wrote down to their expected audience, who they didn't expect to want to watch episodes about invisible enemies guessing where their foes were.
Alternatively, the writers could have been unable to come up with appropriate stories for cloaked ships being blind.
In-universe, it's probably explained in the same vein as the sensors (which can 'see' events light-hours away in real time).
My guess is 'subspace'.
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Sensors positioned outside the cloak would "see" for the ship and wouldn't be detectable in and of themselves as long as they were small passive sensors.
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This is sort of like the invisible man problem.
If the invisible man is totally, 100% invisible, he'd have to be blind.
If he can see, then at least his retinas must attenuate light passing through them, and
his lenses have to refract properly.
Artificial sensors would have similar requirements.
But the thing is, in the case of an invisible man, you've reduced the visible area from a square meter or two, to a square centimeter or two, and in the case of a ship, from (probably) several hundred square meters to (possibly) less than a single square meter.
The cloaking doesn't have to be perfect to be effective.
Of course, if you have a ship that's going to go even just a few percent of C , it should probably be capable of detecting quite small particles, and should therefore be able to detect even the tiny reflection/distortion from the sensors of a cloaked ship.
Of course there's also the possibility of 'active cloaking', that if you know where the enemy you're trying to hide from is, and what their sensor systems are, then you can actively beam just the right signal at the other guy to fool his sensors into not seeing you.
scifi.stackexchange.com/users/96/JustJeff
There is one difference between this question and the invisible man question.
The invisible man issue is purely limited to the visible spectrum of electromagnetic radiation .
For a space ship, it is conceivable that the cloaked ship is hidden in one part of the spectrum (e.g. visible light) but detects other ships using a different part, perhaps X-rays.
scifi.stackexchange.com/users/143/Reinstate Monica - Goodbye SE
Related to  Invisibility should cause blindness: how does hard sf cope? , I'm wondering whether the cloaking device implementation in Star Trek includes an explanation as to why cloaked ships can still observe their surroundings.
How come cloaking devices in Star Trek allow the cloaked ship to continue to observe its surroundings?
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<star-trek><technology><cloaking>
scifi.stackexchange.com/questions/1499
scifi.stackexchange.com/users/559/blueberryfields
The cloaking device is Star Trek is imperfect, as total invisibility would cause issues with seeing. For example, if the invisible man was 100% invisible, he would have to be blind. However, there is also the possibility of 'active cloaking' - if you know where the enemy you are trying to hide from is, and what their sensor systems are, then you can actively beam just the right signal at them to fool their sensors into not seeing you. For a space ship, it is also conceivable that the cloaked ship is hidden in one part of the spectrum (e.g. visible light) but  detects other ships  using a different part, perhaps X-rays. The issue of invisibility and blindness can be dealt with by ensuring the invisibility or claoking is not 100%, allowing enough light to see. Alternatively, sensors can be installed outside the cloak and act as the spaceships eyes. Moreover, the ship can be invisible in one part of the spectrum yet visible in another (visible light versus X-rays). On the other hand, authors may have decided not to deal with the issue as most of the audience would not be aware of it and finding storylines would be more difficult.
The spaceship would only be blind if the cloaking device were 100%. A slight imperfection would allow the ship to "see" Sensors outside the cloak could also be used as eyes.  Active cloaking could allow the ship to continue to see. The shipp could be invisible in the visible light spectrum but still be able to see in the X-ray spectrum The authors decided not to deal with the issue of invisibility and blindness as most of the audience would be unaware and finding storylines would be more difficult.