"You would need at least 512 coins for the first toss. No? "
No. One coin toss for all 512 pairs (or 550 for half of 1100 guests) is probabalistically identical to 512 (or 550) tosses.
For the record, in giving the answer Ray says, “… we take those 512 and we divide them into 256 pairs now. And, of course, we’re going to have 256 people flipping coins.”
And for the record, one coin being flipped for all pairs would also be valid.