Only through chemical means which triggers relatively far away neurons to produce pain. This has the consequence that the pain is not felt at it's source.

E.g. a doctor will become very nervous if you report dizziness with a pain moving from your chest into your left shoulder or neck. Such pain originates along the path of the vagus nerve, which includes your lungs and heart (and half the digestive organs, and even the womb, kidneys and gall bladder sometimes). It's very tricky to diagnose, and usually extremely serious (as in ignoring it may cause death).

You're correct, but I've personally experienced the kind of referred pain you're describing, for both (at different times) prostate and kidneys, and doctors (and now me!) are pretty able to identify that and treat it.

We get plenty of sensations from our internal organs; just ask anyone who had kidney stones. The sensations may or may not feel like pain, but the internal organs are definitely not silent. The issue of referred pain is mostly related to organs sharing nerve supply at the spinal level, e.g., the liver and the left shoulder.