According to the site linked above, chaya usually has stinging hairs and sap that causes skin irritation.The plant that I took clipping off of did not have stinging hairs and the milky sap did not burn my hands (thankfully!). One thing to note that is very important about this plant is the necessity to cook it before eating. The raw leaves contain glucoside that can release toxic cyanide. This would be most unfortunate to eat! To be safe, you must cook the leaves for about 20 minutes. This is similar to cassava.
To propagate the plants, I cut a bunch of sections about 8-10 inches long and pulled off stems that were close to the bottom of each stick. Here is what I started with: