Composed Method: The idea of, rather than having a super long/complex method, you break the functionality up into lots of little methods that are, like, 1-5 lines. Folks general construe that these methods should be private.
Method Object: For a method that would normally take a huge number of arguments or has a lot of local variables that make it hard to do Composed Method, you make an object, store the variables as state and then compose your shit in there. And the original method just delegates over to the new object.
The reason I asked is that it seems like what people generally construe as "should be private" in Composed Method could reasonably be expected to be public on your Method Object. That, anyway, is what came to mind with the phrase "public on another object".