A very important aspect of receiving (and striking) Kote is to remember that the target is actually the heart. It is simply that the receiver attempts to cover their heart with their kote which results in the differently named strike.
As with Omote, Ura and Shita the striker does very little different to a normal strike.
As shown at 00:31 even in the basic exercise the receiver does not open the kote and then command the strike. To do this correctly the receiver gives the command to strike (“Kote wo Tsuke” the first time; “Onaji” for successive strikes) and, as the striker performs the strike, moves the kote into a position where it is blocking the heart from the omote strike. This helps the person striking learn to attack the changed target distance correctly.
The receiver must be careful not to have the kote in an area where it does not cover the heart. If the kote is too low / high (00:40) or too far to the right or left the normal target on the kata is open and is what the striker should hit.
Ideally the movement by the receiver should replicate what happens in shiai (00:37). The striker applies pressure to the omote and the receiver attempts to block the omote strike.
When moving the kote into the blocking / receiving position it is acceptable for the front hand to slide a little and the back hand comes forward slightly (00:50).
Note that the striking surface of the kote will be a lot closer to the striker than the kata would be (the thickness of the wrist basically). As such the receiver will want to “catch” the strike with their wrist and absorb the momentum by bringing the kote back towards their body as they do the customary small step (01:04).
I believe that when learning this strike you should take care not to strike with full power until both receiver and striker have mastered the basic movement. A bad strike to the wrist can be very painful. You will notice that jukendo kote have an extra bit of padding over the base of the thumb compared to kendo kote. If you are using kendo kote this missing protection can be dangerous. Naginata kote are simply out of the question in my opinion.