May be I don't know the Rust detail too well.  Are they the same thing, making the structure priv or making all the fields priv?  Basically I don't want the fields in Inflator to be accessible outside.  Only the methods are exposed as pub.

Reader/Writer for deflate is not a good idea.  The deflate algorithm is pretty low level and it's best to have API operating on buffers instead of Reader/Writer.  e.g.  Decompression might not know how much data to consume and might need to take in extra data beyond the compressed data.  Those data need to return to the caller to handle them.  Reader/writer API make that impossible.