Yep. I have a small bench drill press. Slip the blank onto the barrel trimmer without contacting the blades. Turn on and hold the blank with both hands, using thumb and index finger. Push up and trim. If the blank starts to catch, just let go.
If your barrel trimmer is sharp then you shouldn't need to push very hard.
I never use my whole hand (in a fist) to hold the blank as the top of your hand can contact the blades or drill press, and the bottom of your hand can contact the shaft poking out of the bottom of the blank. Not good either way.
As for sharpening, I use a fine diamond sharpener on the vertical faces by hand. Push the sharpener flat against the vertical face and sharpen upwards (trimmer blades facing up, no shaft inserted). Never try to sharpen the top faces by hand. It is very hard to get the correct angle on those faces and keep all the blade edges the same height.
(Same as the library article referenced by Rick, :redface: oops)
Tony O