Lightroom mobile isn't designed this way, so you can't use it for that. How I sync between computers stems from a want of ease. I just keep all the files and the catalog on a fast external drive that I move between computers. I have tried using Dropbox for the catalog, but it really runs the risk of catalog corruption and I'd had a few conflicted catalog copies, where there have been issues. Even if you don't get corrupted, the conflicted catalogs may contains edits that were done and didn't transfer to the newest version of the catalog.
Would I like Lightroom to do this automatically? Of course, but as LRm uses Smart Previews, rather than original files, I think it's way off at the moment.
Thanks, seanmcfoto. So, just to soothe my sense of insecurity, as an example, I might do some editing. catalog-altering, importing via computer A and those results would be on the attached external drive at that point. Then I could hook up the external drive to computer B and all I have to do is access the newly edited photos, recent imports, etc. on computer B? And vice versa. There is no digital "signal" that attaches the photo library, catalog, etc. to the copy of LR on one computer and not the other, right? Thanks for sticking with me on this.
Once the catalog and the photos are on the external drive, Lightroom will have no trouble. You do need to shut Lightroom down before removing the external drive though. If you were on PC, you'd need to make sure the drive letters were the same for the external drive on both machines, but as you're on Mac, it does a better job identifying the drive.
Right now, I've the large external (a 4TB -G Raid) home from the studio where I'm running the catalog that I use in the studio here at home on my Macbook Air-from the external of course.