I'm not familiar with the particulars of the 350 (I have a 550). It sounds similar to my set up though. A breather tube from the top of the engine to the bottom of the filter box. The filter box has a "pinch tube" that collects gunk which then requires periodic emptying with a simple pinch.
The second chamber, the air plenum, also has a drain. You can look on a parts fiche to see if a pinch tube is needed here too. I have an open tube here to the swingarm. You really want the plenum to drain as any carb overflow that gets in here needs to be wisked away quickly. A flooded plenum could lead to a flooded carb throat which could lead to liquid fuel in the cylinder. If your piston attempts to compress a liquid, it will NOT win!!