Scale allows them to offer better deals -- if Groupon can generate 1,000 takers for a Dreamdu5t Cafe coupon and Google Offers can only come up with 200 then Groupon can drastically undercut whatever margin Google is taking and win your sale.
Scale is THE moat in this arena.
"Yet they haven't shown any way that scaling will reduce costs much."
This is simply Parkinson's Second Law: expenditures rise to match income. Hence the IPO. They need as much cash now as possible in order to scale as fast as possible to build that moat as big as possible to fend off the barbarians.
As the cost of getting more quality customers goes up you'd expect their costs in that area to proportionally decrease. And at that point, if they've defended their dominance in the deal-of-the-day market then they'll become massively profitable else if they're in a decent 2nd place then maybe Microsoft will buy them out for $5 bln more then they're worth, else they'll end up going the circuit city route.