ardly:
Do you not take the Arduino output and feed it to one BC558 and also through an inverter to a second BC558 ?
Would that not end up with the LEDs attached to one BC558 being on when those on the other were off?
Oops!
A BC558 is a PNP transistor - the wrong sort for this application as per septillion's comment.
What are needed are NPN transistors - whatever general purpose ones are rated at least 25 V and a few hundred mA.
Note that each transistor must have a series base resistor, and each LED must have a series current controlling resistor. For a 12 V supply, you can have at least two LEDs in series, 3 is fine for any colour.
It's not a "flip-flop" and it's not an "OR". It is unclear whether the OP wants to control the LEDs by Arduino or just by a switch. To drive some LEDs from an Arduino output, you have a 2k2 resistor feeding the base of the resistor from the Arduino output. the emitter is always connected to ground. The collector goes to the resistor, LEDs and thence the 12 V supply.
Now, the alternate set of LEDs are connected with the same components, but the base resistor to this transistor comes from the collector of the first transistor. The first transistor is the inverter.
Actually, this circuit is more correct:
Ignore the ammeters (comes from a simulation).