"Raj does not live with Aamir."
Translation:राज आमिर के साथ नहीं रहता है।
It's just a structure. This is how I (non mother tongue) have come to think of it:
आमिर के सामने (Aamir's front)
आमिर के पीछे (Aamir's behind)
आमिर के साथ (Aamir's 'entourage/accompaniment/associations')
So the given sentence is like "Raj, (as) Aamir's accompaniment, doesn't live."
By the way, साथ alone can mean together (sometimes you see it like साथ-साथ, or एक साथ). So they go together is वे एक साथ जाते हैं।