The twin paradox and time dilation more generally are widely misunderstood, even by some physicists. The key points to bear in mind are as follows:
If two people start together, move away and then return, their ages will have changed by different amounts upon re-meeting if their respective paths through spacetime have not been entirely symmetrical. In the classic version of the twin paradox, one twin (on Earth) remains in an internal frame throughout while other twin does not, so the arrangement is clearly asymmetric.
At any point when both twins are moving inertially, the effects of SR are entirely symmetrical. For example, on the outbound leg of the travelling twin's journey, the time on the travelling twin's watch falls increasingly behind the local time in the Earth's frame, and the time on the stay-at-home twin's watch falls increasingly behind the local time in the travelling twin's frame. To put it another way, each twin appears time dilated to the other.
The difference in the age of the twins when they re-meet is accounted for by the turn-around event, when the travelling twin shifts from the outbound inertial reference frame to the in-bound one, and it is a consequence of the relativity of simultaneity.
At the turnaround point, the orientation of the plane of simultaneity of the outbound twin rotates with the change of reference frame. The consequence of that rotation is a step-change increase in the time back on Earth which is simultaneous with 'now' on the travelling twin's watch. There are countless diagrams on the Internet that show this if you care to look.
When you ask 'what does each twin see happening to the other?' you must take into account the Doppler effect. That effect will generally cause each twin to see the other as slowed down when they are moving apart and as speeded up when they are moving towards each other. However, there is another important asymmetry here to do with the timing of changes to the value of Doppler effect, which I will now explain.
If you are receiving a light signal, the extent to which it is Doppler shifted can change for one of two independent reasons- either you can accelerate or the source can accelerate. While both will eventually bring about a change in the Doppler shift affecting the received signal, only in the first case is the change immediate. If the source accelerates you will not see any effect until light from the accelerated source has reached you; whereas if you accelerate you will see the frequency change straight away.
If you now apply that to the twin paradox, when the travelling twin accelerates at the turnaround point, they immediately see light from Earth changing from being red shifted to being blue shifted, or, to put it in other words, if they were viewing a clock on Earth through a telescope they would see the clock immediately switch from being slowed down to being speeded up. However, on Earth it takes time for light from the newly accelerated travelling twin to reach Earth, so a person on Earth watching the travelling twin's clock would not see it change from being slowed down to being speeded up until much later.
In terms of what each twin sees, each twin sees the other slowed down by the same amount on the outbound leg, and each twin eventually sees the other speeded up by the same amount when the are moving together. However, the outbound twin sees the switch happen immediately at the half-way turning point, whereas the Earth twin does not see the shift until later. The Earthbound twin therefore sees that the traveller spends more of the journey slowed down than speeded up, while the travelling twin sees the Earth twin slowed down for half the journey and speeded up for the remainder. As a consequence, the travelling twin sees the Earth twin age more.