For hotels there is a third requirement you're not thinking of - exposing yourself to aggregation sites (think Expedia) for bookings.
If this is a requirement (and it probably is when they ask "so how do we sell through Expedia") then synchronisation sort of goes out of the window (there are ways around it but really you all want to look at the same data source).
So single hosted database, single set of web services and then call the web services from both the hotel staff interface (desktop or web) and the customer facing interface. If they ever want to sell through someone else, the web services can be used for that too.
But you should also look at buying this stuff - it's a solved problem and there are off the shelf solutions. Most of them are relatively low on capital outlay, they just take a per booking fee.
(Answering your other question - banks have centrally hosted systems, single data stores accessed live over a WAN - at least they do these days).