Contractually, you just might have to. Even as a technician, a part of my condition of employment was that I had to clear working with another pharmacy chain with the DM. Granted, I didn't really care since this was a temporary gig anyway.
Ethically, I think you should let your supervisors know. Depending on how your workplace is with staffing, getting a 2nd job would/could affect your availability for call-ins and other unforeseen circumstances. By telling your supervisor that you're simply not available vs. that it's a job-related unavailability, they just might give your boss the assumption that you're actually free.
Same goes for things like picking up children or having to take care of an elder parent...boss doesn't have to know details, but it helps them understand why you can't come in.