It can be done but you need to have a very controlled workflow and stick to it meticulously.
Lr stores everything done to photos in its data base, called the "catalog". Without this catalog Lr on computer A "knows" nothing of your edits done on computer "B".
So you need to make sure that when you switch computers and want to continue working that the other computer has access to the latest version of the catalog.
This can be done by
a) copying the catalog to an external hard drive (or an USB stick) before switching computers and then connecting this hard drive (or USB stick) with the second computer.
b) having the catalog on a dedicated external hard drive that is connected to your computers alternatively, so that Lr reads and writes to and from this onecatalog. The disadvantage of this method is that the connection to external hard drives is may be slow which would slow Lr down. Remember that Lr constantly reads and writes to and from the catalog.
In version a) the catalog would be on internal drives of each computer.
Alternatively: You can have Lr write (metadata) changes to file. For Raw files the metadata will be written into an xmp sidecar file. For other files (JPG, TIFF, PSD, DNG) the metadata will be written into (the header of) the original files. By saving to file your edits done on computer A would be available on the computer B even if this computer did not have access to the latest version of the catalog. BUT there is a trade off with the saving-to-file method. Certain things done in Lr (Collections, Virtual Copies, Stacking) are only stored in the catalog and not saved to file. These things would be accessible for computer B only via the latest version of the catalog.
