Get both. A mandrel chuck and mandrels are inexpensive and come in every size. A self centering wood chuck allows you to make long tapered horns. Often both mandrel and wood chuck are used in making a call.
Centering the hole on a wood chuck is difficult. Drill slowly and clean the hole often. Make sure everything is lined up. Get high quality drill bits and keep them sharp.
I use both now, but mostly a mandrel. I do have problems with deflection with the mandrel. That is was brought up my question. I am using a 1/4 pen mandrel. I have a 4 jaw chuck but it is not self centering.
I was on the verge of buying one for $100.00 bucks. But seeing as how Eric has shown me the light I will be getting a collett chuck instead as I already have arbors. Thanks Eric for the redirection.
I use a 4-jaw chuck. I have a mandrel from Hut but never used it. I used to be a machinist so the 4-jaw seemed natural. If I need to put a piece back on the lathe, or if I have a piece of round stock that is close to the finished diameter, I get it "chucked-up" close by eye and then use a dial indicator to make sure its running true.
I did use a 3-jaw self centering mini chuck for a while. It has a #2MT and was not that much $ but it would start working itself loose after drilling or heavy turning.
You can see the indicator in the background of this pic, its on a magnetic base.
I use both. Make the call using a four jaw and finish the mouth piece by using mandrel. I used to grab the bell with the jaw chuck but I got tired of picking my calls off the floor or cracking them. I think Eric has the best idea. Might have to try that.
I use both a Talon chuck and mandrels. If I'm making turkey calls, I use the chuck. If I'm making duck, goose, deer or predator calls,I use mandrels. I don't use a pen turning mandrel. I made my own mandrels from O-1 Drill Rod. Brian.