The first thing to do is to try to identify the source of the noise. I'd leave on computer off entirely, and focus on the other. Start by eliminating all peripherals -- disconnect the monitor, and leave it powered down, remove the mouse, keyboard, and any other devices - including ethernet switches or routers, cable modems, or telephone modems -- and unplug anything connected to AC power. Remove all cables from the PC except the power cable. Now turn it on and let it boot.
If you still hear noise on your transceiver, then the problem lies within the PC, and is likely being radiated through the power cord. Try wrapping the power cord around some appropriate ferrite cores. If the computer's case is metallic, try grounding it. Also, look for poor electrical connectivity between the two halves of the case.
If you don't hear the noise with all peripherals disconnected and powered down, then add them back one at a time, rebooting after each addition. This procedure will identify the culprit, but keep in mind that there could be more than one. Correcting the culprit will depend on what it is, but wrapping its interface cable around some ferrite cores would be a good first step.
Once you quiet down computer A, apply the same fixes to computer B; if its still noisy, repeat the process described above.