Hey John,
I usually cut my caulerpa only one end at a time because its cell vacuole can kind of spill out at wounds. It is less stressful on the plant to just cut it in one place at a time. If the plant gets really stressed out it can die and release all the nutrients it has stored up in the form of gametes and dead cellular junk as well as the anti-herbivore toxins it makes. Chaetomorpha is a filament of many separate cells. I just grab a clump of mine and pull it in two directions when i need to split it up.
HTH,
Kevin