What do you mean by "modules that are independently compiled"?

If I were to write a kernel module, which I only distribute in source form, there is nothing that would require me to make my module GPL unless I have copied into my module source copyrightable elements from Linux (or some other GPL-licensed code). Let's assume I use a license for my module source that is not compatible with GPLv2.

If you take my module, and compile it into a Linux kernel, and want to distribute the resulting binary, then YOU have a problem in that the resulting binary that includes the kernel and my module is probably a derivative work of the Linux kernel, and the license of the kernel doesn't let you distribute derivative works of the kernel unless they are licensed under GPLv2. (You might try to argue that my module and the kernel together are a compilation, not a derivative work, but I don't think that would work).

I've seen it argued that since my module is intended to be built into kernels and doesn't have much use if not built into kernels, than I am effectively building it into kernels and distributing it even though others do the actual building, and so I do have to license my code under GPL.

Copyright law (at least in the US) does actually recognize a couple of circumstances under which I could be liable for someone else's infringement. There is "contributory infringement" and "induced infringement". If either of these could be applied, I could be held responsible for the infringement that occurs when others build infringing kernels using my module source. For induced infringement, the person actually doing the infringement basically has to be an agent of mine, so that would not apply. For contributory infringement, I just have to be doing something that enables their infringement.

Alas for those who want to hold me liable, there is a major problem with the contributory infringement approach. There CANNOT be contributory infringement without there being a direct infringement for me to have been contributory toward, and there must no substantial non-infringing use for my work. As even Stallman concedes, the GPL allows a person to take GPL source code and combine it with code that is licensed incompatibly with GPL and build binaries for their own private use. These people are not infringing, and so do not provide a foundation for contributory infringement, and they provide an example of a substantial non-infringing use for my work.

The above hypothetical (module distributed in source form only) is what I think of when I hear the phrase "independently compiled".

One thing often overlooked when these discussions come up is that the Linux kernel is not the first program that outsiders have wanted to write code for without following the rules set forth by the program's owners. People have been figuring out non-public internal interfaces and data structures of word processors, spreadsheets, video games, embedded systems, proprietary operating systems, and so on, and then releasing things that patch in and do unauthorized things long before Linux came along. All that's different with Linux is that (1) they can look at the Linux code so it is a lot easier to figure out non-public internal interfaces and data structures, and (2) GPL saves them from contributory infringement (with unauthorized modifications to proprietary programs, the end user might be violating a EULA if they use your work, which puts you on less firm ground).