Some implementations use a global "require", while others have distinct "require" functions. If you are writing portable code, you cannot guarantee that monkey patching a property on require will be reflected in other modules. If you are writing an implementation, you can encourage folks to write portable code by guaranteeing a unique "require" function in every module.
"require.paths" was always stop-gap today is an anti-pattern. Narwhal discouraged its use at run-time and would initialize it using a topological sort of the transitive dependencies. Node.js does not have it at all, and goes on to mandate that top-level identifiers must reach into dependencies, requiring relative identifiers for modules inside the same package.
If you do support "require.paths", I recommend copying it as a property of each module’s "require" function. That will satisfy all of the requirements of the spec. That is, replacing "require.paths" on one module’s "require" will have no effect, but changes to the content of any "require.paths" will, as in "require.paths.push(path)".