Yeah, that's what I figured. iCloud is the only connection between the two. I logged in to iCloud in a browser figuring they would show up there somewhere but they don't. Guess it's just one of those "hidden features". Pretty nice that it does that though.Voicemails likely store in your iCloud profile.
Note: I don't use Apple products, but my wife does. I'm sure that's where I am remembering this from.