"Do you know where the director is? - No, I do not know."
Translation:Ви знаєте, де директор? - Ні, я не знаю.
Is the personal pronoun actually required for a grammatically correct sentence? Can I just say: "ні, не знаю," or does the я have to be there?
Yes, you can just say "Ні, не знаю", this is more spoken and lazy way. I would say it like this in 95% of times definitely.