Originally they told us quite a bit of logic was done on their servers. But then they announced that Diablo 3 would be playable offline on the PlayStation 4. Now I don't know what to believe. Could it be it was all just talk to stop us from trying?
No, item drops are actually done on a server, to guarantee fair drops for everyone(you can sell them for real money, so you would rather make sure nobody just adds an item worth $$$ to their inventory).
PS4 won't be connected to the Auction House, so most likely it will include just its own, internal version of the server that will generate item drops.
The servers are almost certainly running from the same source base as the client. Swapping out who-does-what, compared to a console port, is trivial.
Or that the PS4 version is the first variant that has enough functionality built in for it to be playable offline.
they're probably just running the server code on the ps4 alongside the client. it's easier to secure there