For Boxcryptor (Classic), the only requirement is, that the encrypted files and folders are accessible in the Windows file system (e.g. you can see them in Windows Explorer). The encrypted files and folders do not necessarily need to be located on your hard-drive - as mentioned, WebDAV could be one way to access them without downloading everything.
If you have filename encryption enabled and can't "remotely" map the files to your file system, you would need to download the whole Boxcryptor Classic Folder in order to locate the file you need, yes.
PS: I know it doesn't help you on Livedrive, but we also offer a plugin for Google Chrome where you can directly browse and access the encrypted files on the websites of Dropbox and Google Drive. More information: https://www.boxcryptor.com/en/labs