Probably neither is very "constant".
Simple stick welders are nothing more than a transformer with an adjustable mechanical shunt which limits the energy transfer from primary to secondary side.
With this shunt type power limiting the output voltage collapses once you go over the adjustable current, but the current is also not very stable, and changes with arc length, drops of metal temporarily shorting etc.
MIG welders often have a bunch of fancy electronics in them which can do all sort of fancy things to your welding current / voltage.
The algorithm is probably dependent on the brand of welder used.
For simplicity, let's assume that the impedance of the arc is lineair with it's length.
If you have a CV regulator, then the energy input would get less with a longer arc.
If you have a CC regulator, then the energy input would increase with a longer arc.
My guess is that if you do something in between, you could keep the power input almost constant with a varying arc length, which makes welding probably a bit easier. But making the energy input dependent on arc lenth gives the welder more control during the weld.
Every now and then you find some reference of stick welding with a few car batteries.
This is sort of Constant Voltage, but with welding currents the voltage from a car battery will drop so it is not really "constant" anymore.