'To hand someone something' is a bit more formal. A teacher would say "I will give you a copy now". Using 'to hand sb smth' is odd as it is hard to think when it is used. However: -in a past tense narrative it is often used. Like in a novel: "the man handed the waiter two notes and left" -in quite formal situations, perhaps as a customer, you may be asked to hand sb smth: if you go to a bank for a personal meeting the bank employee may say- "if you could hand me the relevant documents we will get started". But to give is always appropriate.

"Hand a copy" makes sense, but I've heard "hand out a copy" more often.