I have never much cared about self intersecting meshes when slicing with Cura. Geometry like the one below are often practical. I for instance add lots of rivets that self intersect with the base geometry:
Now I have switched printer, and am using Simplify3D instead. All of a sudden, I get lots of problems with these models. The intersection of the objects become hollow. Simplify3D has a setting to join the outer solid shell but it also fills holes (such as the center hole of a gear).
I make models to sell so this is a big deal for me.
EDIT: Also, they printed perfectly fine in formlabs "slicer".
You definitely need to get rid of the self-intersecting meshes in order to be able to 3D print your object. I would personally recommend you try MakePrintable: automatically clean these type of meshes.
Makeprintable is a cloud service (your model is uploaded to their servers and processed there) currently in beta. You need to sign up but it is free (and they state that they will keep it free).