Cloud and hosted are often used interchangeably, but they are different.
Hosted is similar to a local install, except it is done on a remote server controlled by a third party. Typically, you use terminal services to connect and open the application(s). Remote Desktop is commonly used as the connection method.
Cloud is web-based software. Think NetSuite or Quickbooks Online. Even the Creative Solutions suite. Those are all true "cloud" solutions, which you access via a web browser and they are hosted, generally, by the actual software vendor via web servers.
Some local install applications do NOT play well with terminal services, virtualization, etc. Quickbooks is a prime example. Sure, technically it can be hosted and use terminal services, but stability is going to suffer.
The multi-office firms I am aware of either use true cloud solutions, or they connect to their remote servers over a VPN. A specific firm I am thinking of has a local office, but they use VPN to their headquarters' network located 2.5 hours away and/or use cloud software. Another firm I am aware of recently consolidated into a single office after the misery of running multiple LANs and redundant software licenses, with zero connectivity between offices.