I've noticed that these two-handed things started to show up more in Chick Corea and Herbie Hancock's playing after they had done some of their fusion/funk stuff. Clavinets in particular sound really good with this rhythmic stabbing effect.
Anyway, in the first clip, Hancock is playing octave D to C in the right hand and playing two other notes in between with the left hand. So the first beat of the triplet is the RH, the next two of the triplet are the LH down an octave from the RH pinky.
The Chris Botti clip uses a Fourth+2nd below hand shape and two notes in the left hand coming up. For example (he changes notes often, so I'll just show you one of them):
RH: C+F-B LH: B-C RH: C+F-B LH B-C
This one is in sixteenth notes, whereas the Hancock one was triplet. The F is on top of the C.
You'll notice that the RH part is basically a broken chord. That fourth+second is pretty common, though usually it's fourth+whole step below, not half step, but whatever. The LH is just single notes, but he could've used another fourth+second shape to respond with the other two notes of the sixteenth grouping.
So, once these are comfortable, try coming up with your own! Fourths are a good play to start. Grab G-C-F in your RH and LH an octave below the right. Then do like Chris Botti did:
RH: C+F-G LH G+C-F...
Play the fourth first for each hand and then the other note from the chord quickly afterwards. It sounds good if the RH goes down and the LH comes up, but it doesn't have to be like that. Try starting with the LH going up and then RH coming down. Or both hands going down. Use your imagination! Then try different chord shapes. A "power chord" (1-5-1) is also a good candidate. Ex. C-G-C.
Besides these perfect interval voicings, these things also work well for diminished scale voicings. One that my teacher likes to use is two P4ths a half step apart. Ex. C-F-Gb-B. Try C-F-B-Gb, C-F-B-Gb split between the hands. Then sequence it in minor thirds: Ex. C-F-B-Gb, Eb-Ab-D-A, Gb-Db-G-D...
There are lots of possibilities, but the fun is finding the ones you like. My only caution with these super fast lines is that you have to stay in the groove still! It's easy to fudge the tempo, but try to stay exactly on it! With a little practice, it should be no problem.