Yes, they can have separate power supplies but you may need a common ground.
A regular (non-addressable) LED strip can be controlled with a MOSFET or a relay (3 for an RGB strip). With relays you don't need a common ground and you can't dim with a relay.
Addressable LED strips don't need either. They have a built-in driver chip for each LED so they just need power and a low-current data signal from the Arduino. These also need a common ground (as a ground-reference for the data signal).