You would need a CNC Copy Lathe, but s far as I know, they're all much larger and far more expensive than would be needed to make objects as small a drumsticks. Small objects can be turned on a large lathe, but of course large objects can't be turned on a small one. The CNC software needs to be programmed to create the desired shape of the object, and the lathe needs to be operated.
A CNC woodturning lathe would be quite a large capital investment for producing l.ow value items (compared to say newel posts)
Here's an example of a CNC Copy Lathe in action:
Woodturning CNC and Copy Lathes
A much cheaper, simpler, and I suspect easier lathe to operate to achieve the same end result - lots of identical drumsticks - would be a non-CNC 'Copy Lathe'. Such a lathe copies the profile of a sample object - in your case a drumstick - which is mounted between centres at the rear of the lathe and remains stationary during the turning process. The wooden blank is mounted between centres on the lathe, which has to be manually advanced from right to left as it shapes the object, calling for limited skills beyond parting of parting off each object.
Obviously much depends on whether you wish to make drumsticks in such high volume to justify the capital investment in a CNC lathe rather than a simpler non-CNC 'copy lathe'. You might like to Google 'copy lathes' and see what comes up. The other option would be to buy the drumsticks in from a high volume production turning company, but that's not answering your specific question.
It may be that other forum members know of smaller, cheaper CNC lathes, but I hope these notes are of some help.