There is no real "Syncing" options between catalogs.
To keep it short, the smart solution which has always worked is not to sync, it's to share.
You simply store both your catalog and your photo files folder tree on an external drive which you plug in alternatively to different computers with the same Elements version.
You need to create a full backup on an external drive.
You need to restore the backup to the final external drive. That means you have an exact copy of your catalog and media folder tree on the external drive. You can forget the original one, which you may consider as a third backup or completely delete to save space.
The catalog folder is now on 'custom location' just under the master folder you have chosen for the restore.
So, from each different computer, you take care that the external drive is recognized with the same name/drive letter. Once you have browsed to the new location, the catalog will automatically find the custom location.
Is it safe and fast?
If you consider the advantage of the external drive (physical size, great capacity), the very small loss in speed with a USB3 connection will be negligible.So, you only need the time of a backup and restore, which is necessary any way. Of course, you keep your backup schedules as before to be safe. It's easy to transfer such backups to the internal drive of a new computer. Just keep in mind that the catalog won't be automatically stored in the default (hidden) location of catalogs. If you want to, you can use the catalog manager to move the catalog to the default location.
You haven't stated which version of Elements you are using. If you are using a recent version that generates auto-creations, there is an exception to Michel's advice. The files necessary to generate the auto-creations are located within the system folders of each machine, and as far as I know, there is no way to move them. Consequently, there will probably be different sets of these auto-creations that are generated on each machine. You can of course save the auto-creations to a file location of your choosing. But the ACs that appear on the welcome screen will not be synched.
That's interesting Greg. I did not know about the possible issue with auto-creations which I don't use. Since I am storing my catalog and library on an external drive (regularly cloned as well as backed up from PSE) I can say that sharing between my two desktops in two different places works perfectly with my PSE2019.
I imagine that auto-creations can be stored or moved where you want? In that case that would be the same as sharing manual creations like slideshows, videos or books. I use Dropbox or OneDrive to share such files without using the external drive.