I can provide you the solution what I do for using lightroom on dual computer.
I have my catalog and all the images on my external hard drive.
I take this hard drive everywhere and just put it in the machine where Lightroom is installed.
i open my catalog from the hard drive and start editing.
Everything stays there.
If you frequently switch between computers, probably the external drive method works best. You can keep the originals on the drive too, or just smart previews.
Dropbox works fine, if you give it time to sync up from the computer you have just stopped using, and then down to the other computer before you open the catalogue.
You also have the Export as Catalog / Import from Another Catalog method if you want to take a group of images into the other computer.
A light alternative is to use Lightroom Mobile but through a browser. This gives you the ability to review, flag and add star ratings, and also do some broad adjustments.
I use two MBP's and have the catalog on a LaCie Thunderbolt drive with previews and smart previews (make sure Catalog settings are stored with Catalog in Preferences) and image files that I am using currently on a USB3 Drive..... this is fast enough as Lightroom isn't accessing the files so much. My MBP's have SSD drives so Lightroom runs well....
This allows me complete portability, even to plug into any other computer with Lightroom and everything is always up to date...
I was thinking of having, lets say, LR Ircat Catalog file and smart previews synced through Dropbox. My aim mainly is selecting pictures, sorting, renaming, tagging, face tagging, making and editing albums. In this way, I do not have to be around with a drive with my pictures, which could be lost in the way. I want to keep my files in my iMac while being able to do some work on the way.
I think it should work, meanwhile I will just make a catalog backup before I try this way