I think we'll see both, with a sort of push to the two extremes initially, and then the two extremes coming back together to meet in the middle again as WASM gains popularity on both the client and the server leading to a lot more "isomorphic" (stupid term in this context, but whatever) codebases.
I think the eventual common use for WASM will eventually be DRM and related types of page code "security" and obfuscation. Basically a way for companies to "protect" their tech/IP from easy duplication while not having to carry the expensive burden of the cost of back-end processing in the cloud.