Then when you view those photos on another computer, they scan those sidecar files and show you your photos as you intended. Those sidecar files get copied and synchronized along with the photos, using any of the methods mentioned above. Instead you can enable the sidecar feature which creates tiny sidecar files for each photo that you change metadata or make edits and adjustments to. There is no catalog to have to move around. In order to share the photo and its settings you have to move that catalog from computer to computer, often dealing with broken links. That means the photo will only look as intended on that one computer. Most raw editors store all the non-destructive edits and adjustments as well as metadata in a catalog file living on one computer. You just take your photos with you and connect the drive to whichever computer you want to work on. Of course there is the tried and true external hard drive as well. These are common place in work environments. Many photographers who work in studios use these to store there photos on a private internal network where the photos can be accessed from multiple computers. You can also use a file server or NAS drive. It gives you an online backup you can access anywhere and automatically manages keeping your photos in sync across multiple devices.
Same thing if you have Amazon Prime, you get unlimited free Amazon Cloud Drive storage. If you have an Office 365 account they give you 1TB of OneDrive for free. I use Dropbox but there are lots of other great and inexpensive cloud syncing solutions (Google Drive, Microsoft OneDrive, etc). There are lots of ways to get your photos on both computers. I also have a laptop I use when traveling, or when I want to sit on the couch. I have an iMac I use at home for my main editing. Most of us have more than one computer we work on our photos with.