Hi Budoka9,
Within the Bujinkan it is not "trained" as such, but is a skill that is supposed to be developed over time, with repeated exposure to a potentially dangerous environment you become more and more sensitive to the changing energies around. The big issue with this, of course, is that it relies on your training partners/attackers/uke attacking with a realistic intention, allowing you to "pick it up", o rbecome more attuned to it over time. If that intention is lacking (see the Bujinkan's "Happy Heart" training of the 90's), then it is much harder, if not impossible to realistically develop that sensitivity without other training means. This is one reason why the sakki tests of the 80's are quite different to those of the 90's and later, and could be a reason why you are seeing a difference between what you have seen of the Bujinkan and the Genbukan. For the record, the Bujinkan's tactic of having little to no real standards, leaving it up to the individual teachers to pass on what they think is important in a way that they want, means that here, more than anywhere in the Ninjutsu world, it will depend on the instructor more than the organisation. You simply cannot judge the entire Bujinkan on any one individual, unless that individual happens to be Hatsumi Sensei.
As to the Genbukan and Jinenkan, I have not had the exposure to their teaching in this regard, but I assume the idea of developing the skills will be fairly similar. But the higher focus on standards will mean a more even representation across the organisations. But in my organisation, we actually do train the skill specifically, utilising visualisation, and simple training exercises, focusing on the transmission of a sakki intention.
As to the way the specific kata are trained, whether the Togakure Ryu ones or the Gyokko Ryu ones (or the kata from any of a number of other schools, or in Iaido, or as a henka of really any technique at all), they are often trained with the uke giving a soft kiai before the attack, allowing the defender to recognise the intention and react to the attack. Later in training, the kiai is left out, with the defender relying on their sensitivity to keep them safe. When this happens depends on the student and the instructor.
Hope this helps.