We propose and study a nonlocal system of balance laws, which models the traffic dynamics on a two-lane and two-way road where drivers have a preferred lane (the lane on their right) and the other one is used only for overtaking. In this model, the convective part is intended to describe the intralane dynamics of vehicles: the flux function includes local and nonlocal terms, namely, the velocity function in each lane depends locally on the density of the class of vehicles traveling on their preferred lane and in a nonlocal form on the density of the class of vehicles overtaking in the opposite direction. The source terms are intended to describe the coupling between the two lanes: the overtaking and return criteria depend on weighted means of the downstream traffic density of the class of vehicles traveling in their preferred lane and of the class of vehicles traveling in the opposite direction on the same lane. We construct approximate solutions using a finite volume scheme and we prove existence of weak solutions by means of compactness estimates. We also show some numerical simulations to describe the behaviour of the numerical solutions in different situations and to illustrate some features of model.