Can't you? You ever get a cell phone bill? They usually read like telephone books, nary a call made, even if its for 1 second, is omitted.
What is stored locally would be a phone directory and the actual text messages, perhaps, but that would be erased only if the "hard drive" was removed, I believe, not a "dead battery"
Yes the billing system would save the calls completed, but not necessarily the calls not answered. I don't know.
The "local" information could be interesting also.