So if someone wanted to serve their own bitcoins, would they have to run separate client and server software?
No, the "server" version would be standalone as it is now.
I would think that a blockchain server would be more useful. This would allow phone clients to have a local wallet.dat, and query a blockchain service to serve up the blocks necessary to validate a receiving transaction without needing a local copy of the blockchain. A network drive might be able to do this decently well, but would result in a lot of network traffic for a smartphone.
Well, why not have both wallet and blockchain servers and let the user pick.