Answer
The magnitude of the disk's angular momentum and the magnitude of the disk's angular velocity both decrease.
Work Step by Step
We can write a general expression for the angular momentum:
$L = I~\omega$
Since the beetle walks in the direction of rotation, the magnitude of the beetle's angular velocity $\omega$ will increase, which in turn increases the magnitude of the beetle's angular momentum.
Since there is no external torque on the system, the angular momentum of the beetle-disk system remains the same.
Therefore, the magnitude of the disk's angular momentum must decrease, which means that the magnitude of the disk's angular velocity must also decrease.
