3-handed siteswaps are generated mostly the same way as 4-handed ones - just 1 hand always has a "0". As 0 is an empty hand in siteswap, the hand with always 0 won't do anything in the pattern.
At the Israeli juggling convention in 2022, we had a very sad reason to develop the idea of 3-handed siteswaps into a workshop. On the first day, just before the first passing workshop, Ofek hurt his right shoulder so severely he had to carry his arm in a sling for several weeks to come.
There are many more possible 3-handed patterns - we just picked a few interesting ones for this workshop.
Pattern #1 6860-6770-6770
Pattern #2: 6860-6770-8570
This pattern is interesting for the 1-handed person and easy for the 2-handed person.
Pattern #3: 7706-7706-8804
Pattern #4: 5780-5860-6780
This one is hard for the 2-handed person (A). All Passes and Zaps are crossing!
Pattern #5: 8570-6860-8570
All Passes and Zaps are straight.
Pattern #6: 9605-7709-5606 "Ofek"
Here, the 1-handed person practises their 1 hand on "holy grail", doing Doublepass, Singlepass, Zap.