When you think about rain in the distance (generally), you will have rain at several different depths from the camera, some close up, some very far away, all of which will look slightly different as you won't be able to focus on all of them. But the effect of them all layering on top of each other as they go off into the distance is what helps give the look you're aiming for, you need to recreate that depth and layering to really give the look of rain further in the distance.
Particle emitters are the general method of creating rain, as usually they use very small meshes, or even just a texture sprite of a drop. This keeps them pretty efficient - to a degree as the more particles you use and have on-screen at a time will change how heavy or resource dependant the effect becomes.
Generally you would only use large detailed meshes for rain, when you have a close up of a puddle and wanted to animate the splash, or even a close up of the single drop for some reason.
This can be combined with distance fog to further increase the sense of depth in your scene by occluding the objects in the far distance.
Again you could use another particle emitter to create dark, heavy clouds that go off into the distance, as having clouds in the scene will help to sell the look.
A lot of this will depend on what software you're using, as well as its intended platform, as different software have different solutions but particles and fog are pretty commonplace.
tldr: Particle Effects & Distance Fog
