What are the pros and cons of having a side-stick versus a centered control stick in a fighter aircraft?
Center mounted sticks are useful for maintaining positive control of the aircraft while operation some cockpit system on the right consoles and flying with your left hand.  It’s also useful in an emergency, say if your right arm was injured by shrapnel, etc and cannot actuate the stick effectively, requiring you to fly with your left hand.

Side sticks do have the advantage of less cockpit clutter.  The stick is mounted off to the right and doesn't directly block critical instruments from the pilot's view or have its range of motion restricted by flight gear, kneeboards, etc.

As far as size and weight are concerned, both center mounted and side sticks are pretty much the same in a fly-by-wire aircraft - just a 4-way transducer LRU with a handgrip attached to it.  In a/c with conventional control systems the center stock was a necessity as it acted as a lever arm allowing the pilot to overcome aerodynamic forces on the control surfaces with muscle power alone and one could force a maneuver using both hands.  In the age of hydraulic flight controls and fly-by-wire, this is irrelevant.

As to ergonomics, it's fairly comfortable to fly a center stick as your wrists will naturally rest on your thighs when seated and your hands naturally float to that position.  Side sticks are also comfortable but require an armrest to operate effectively, particularly when pulling Gs.