Meeting a Girl in Saudi Arabia
You can be brave and take risks, many people do when dating girls in Saudi Arabia! It is actually quite rare for a westerner to be stopped and challenged about the woman he is with, although it does happen. But be sensible, you are a westerner, being accompanied by a fully veiled woman will raise a few suspicions, best for whoever you take out on a date in KSA to behave in a western manner even if not western, so no veils and uncovered hair.
A westerner walking with a Filipino or Indonesian girl will attract quite a few stares, especially in Riyadh, which is far more conservative than other cities. This can be uncomfortable for the woman and there is the potential for someone to say something. Many Arabic women look quite western so there are fewer stares, but they are harder to date (I'm told!)
Going out to dinner or a walk around the mall and a quiet, romantic coffee can be achieved, but little else, getting a room is quite difficult, requiring your ID and so on! The hotel is not going to turn a blind eye to you bringing a woman to your room I promise you! I do know one hotel in Riyadh where you can get from the parking to the rooms without being seen going through the lobby - if you need to know just message me.

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