When I move the EndNote library between two machines, the groups / attachments disappear
This is usually because you have not included the entire EndNote library. Such a library consists of a file (e.g. «library.enl»), and a directory structure (e.g. «library.Data») with additional files. These must be in the same place to be used jointly. If you do not copy with the folder structure, groups / attachments will not follow.
The easiest way to make sure you include the entire EndNote library is to use the «Compressed Library» feature.
(See more on backup and copying the library.)