Although, in fairness the "generous people" on whatsyourprice don't have to pay on-line as they do with eBay. It sounds like a conventional escort transaction from what I read on the site.
If a police department with enough time on their hands get the idea that money is being exchanged for sex via that site, they'll just list themselves on it and wait for someone to either ask for or give them money for sex and then arrest both that person and the site owners immediately. That'll be it and everyone will be pouty.
It seems to happen periodically with US directory/review sites - not
that often because, like here, it can often depend on the political climate whether or not a whole police department will pour resources into arresting harmless consenting adults who want to pay/get paid for sex (if there as a whiff of underage prostitution or trafficking around a site then there's usually an immediate national focus on it, of course - isn't that what happened with Craiglist's Erotic Services section in the US?) ... it also depends on whether the site is hosted in the US and whether the site's owners are US residents. When American directories go down it can be
extremely messy for all the members - especially if the owner starts telling the police who everyone else really is in order to reduce their own chances of jail time. Not so much industry solidarity at that point!
I believe AW's owners are based somewhere in Spain? But they operate mostly here, so although those guys basically spend all day long gaining from the control (or at least, encouragement!) of prostitution, it's unlikely that there'll be a quiet enough week crime-wise for the police to start faffing about with what, again, mostly involves consenting adults doing things in private anyway. The AW push for verification (trying to get all their members to send in scanned photographic ID) is most likely to do with protecting themselves against allegations of being involved in trafficking and/or child prostitution - those are the main things that any prostitution-related site (or, really, any adult site) needs to avoid like the plague if they don't want their hosting company to immediately take their site down while simultaneously giving the police all of the site owners' stored billing information.
At the end of the day, I think any member of any police force anywhere in the world knows - like most other sane adults do - that prostitution has always existed and probably always will (til they invent proper sex robots, I guess?! Hopefully I'll be long gone by the time that particular creepy event happens!) so making it and/or related stuff illegal not only makes it all the more dangerous for those involved but is also a colossal waste of police time and government money. At least ignoring things that aren't causing anyone any real bother helps cut down that waste. Putting a stop to child prostitution or trafficking (where it genuinely exists) is very important, of course, but hassling private consenting adults while making it easier for scammers and worse to find victims who will be less likely to report them is not so good.
I'd expect that anyone making an adult site these days, particularly if related to escorting, could probably expect a 60% chance of being shut down by anxious hosts or disapproving complainers (or angry rivals), a 6% chance of making a bunch of money and a 34% chance of being totally ignored.