I’ve worked with rain footage with green screen and I can tell you, it’s a nightmare. Shooting everything separately and adding the rain in post-production was the way to go for our experience. If you’re not involving green screen, shooting rain on set will give you the most realistic effect. No Film School put together a great article on how to shoot rain scenes correctly.
Create Rain in Post Production
If you need to add rain during post-production because of green screen, stock footage, or whatever other reason, it’s not as difficult as you might think! There are lots of methods of adding rain, including plug-ins and stock footage.
Michele from Toolfarm wrote a gigantic article, In Depth: Rain Generators which has been recently updated. The article talks about the following plug-ins and particle generators, plus some tutorials.