Really depends where it's leaking from. I know you have added you pictures but it doesn't really tell us if it's leaking through the waste/overflow section at the black nut and leaking at the trap connection as well as the overflow pipe connection.
1)There may not be enough thread there to allow the trap to tighten up enough so the seal can compress and become water tight. Judging by the tool marks on the white plastic nut I'd say that's one issue, you shouldn't need tools to tighten that nut for it to be water tight. Once tightened up can you still move the trap side to side?
2)the overflow pipe should self seal, unless the pipe itself has expanded too much
3)the waste section itself should have rubber seals that make that watertight.
I'd probably change that waste to a new single piece overflow type.