Some of the editing in LR can be done using the preview files. Some of the editing requires access to the original files. If the original files are needed, and if they are not available (like on a disconnected drive), then LR will give you an appropriate message and not complete the task.
If you regularly want to edit images that are often disconnected, you can turn on the "smart previews" and pay the price of used disk space for the convenience. In many cases, but not all cases, the smart preview will cover many of the needs for the original file.
You image files can be located practically any where: internal drive, external drive, network drive. Sharing the images between to computers is relatively easy.
Your LR catalog needs to be on a "local" drive (internal or external). "Sharing" it between computers takes care.
The easiest way (not prettiest way) to share your catalog between computers is to put the catalog (and the images) on an external drive and move the drive from computer to computer as needed.
Thanks guys. I'm beginning to be swayed towards the external hard drive solution. Dropbox gives me 2Gb free, but my latest catalog (which contains information about all my images, including previews) is 10 Gb. I'm thinking it will just be simpler to use the hard drive containing catalog and images, as long as I back it up. Thanks again