Thank you all very much for your clarifications and musings. (Actually, Vanadium, hearing from the similarly "confused" is helpful, or at least interesting.)
I like your explanation, Drakkith. Almost seems to say the math is defining more than describing reality. It jives with what I've read since posting my admittedly unenlightened question: that virtual particles aren't particles at all, but disturbances in fields. As Matt Strassler puts it: "A particle is a nice, regular ripple in a field, one that can travel smoothly and effortlessly through space, like a clear tone of a bell moving through the air. A 'virtual particle', generally, is a disturbance in a field that will never be found on its own, but instead is something that is caused by the presence of other particles, often of other fields." Note that in his simile he likens them to compression waves.
Intrigued by your rhetorical parody of my question, rumborak. Coulumb force/interaction is the attraction or repulsion (not emission) of particles. I guess your analogy is lost on me a little.