I doubt that US rules would be different, in Australia you must have two way communications with ATC or a Flight Information Service at all stages of flight. Once out of VHF radio range (a little more than “line of sight”) then you switch to HF which has very long ranges. There are also satelite communication options.
But there’s a difference between “radio contact” and “radar servcies”. Sure, you can always call someone, but that doesn’t mean they’ll be giving you flight clearances. I don’t know where, exactly “radar services” terminate but I’m pretty darn sure they don’t extend across the whole of the Atlantic, much less the Pacific.
But that’s quite nitpicky.
Well radar services doesn’t equal controlled airspace either. You can get a clearance to enter controlled airspace without having to talk to the actual controller, and you get controlled airspace without radar, sometimes you need to get a clearance relayed through a flight service dude on HF.
Of course Australia is very differenct than the US in that most of it does not have low/mid level radar coverage.
You can listen to ATC from several parts of the country on the web now too. I tune in sometimes and it never ceases to amaze me how air traffic controllers can do that job when air traffic is busy. That is one very difficult, high stakes video game that never ends.