Woodturning tools are made from high speed steel (well the good one’s are) screwdrivers are not, as Woody has said some have harden tips so it possible to just grind the tip. The problem however is that the screwdriver could/will bend which will weaken the steel and it will break.
You could temper the steel as again Woody has said but how do know when the steel is up to HSS standard.
If you want to make your own tool and know it will be safe to use I would buy a piece of HSS rod and shape that, but bear in mind while profiling the steel you do not want to get it above hand hot otherwise you have un-tempered the steel.
If you really want to go all the way then Woody’s link will help.