> How do you get both sides to have the same infinite random pad?
With Quantum Key Distribution both sides have access to identical streams of random bits not available to anyone else.
Without invoking quantum mechanics, you can just treat the portion of the pad you have on hand as a prefix of an infinitely long pad which is being delivered incrementally. If you run out of pad bits you just pause and wait for someone to hand-deliver a new set of identical storage devices to both parties with the next segment of the pad. This is the same principle behind treating physical computers as "Turing complete" when a Turing machine is technically defined to include an infinitely-long tape—you can always extend the system with more storage as needed.
I thought the whole problem with one time pads was that they had to be distributed to the other side physically and thus were always finite.