Do you know where your circuit breaker panel is (or perhaps fuses...)?
If you are lucky, the panel may list the rooms/outlets protected by each circuit breaker.
Otherwise, you will need to do one of two things:
1) Plug a lamp into each of the candidate outlets. Turn off the circuit breakers one by one. If both lamps go off with one breaker then the lamps are on the same circuit.
That method is safe but you may end up turning off devices that will need to be reset afterwards - just as if your power went off.
2) The other method is to purchase a test set that uses a tone generator. You plug the tone generator into an outlet and scan the breakers with the detector.
When the applicable circuit breaker for that outlet is scanned...