To start, create a new catalog in your dropbox folder. If you are using the free 2Gb account then I suggest you use minimal previews on import. Also, in Catalog Settings set the 1:1 previews to be discarded after 1 day.
The reason for this is that 1:1 previews can grow the previews database to a rather large size. With only 2Gb to work with that can become a problem quickly. Of course, if you have a 50Gb or 100Gb paid account this might not be a problem for you.
On computer #1 everything works as expected. When you close Lightroom and go to computer #2, open the catalog from your dropbox folder on that computer. If the drive the images are stored on is not yet mounted on computer #2 you will see the missing folder/files indicators in the Library module.
After mounting the drive with the required folders and files then Lightroom is ready to access your original images. Since the catalog is exactly the same as the catalog on computer #1 the names and locations should be correct.
Of course, since Lightroom is not a network enabled application be sure and close the catalog on computer #1 before you open it on computer #2. Lightroom does create a .lock file when it opens a catalog but it is better to be safe.
Theoretically, Lightroom uses the same database structure on Windows and Mac. However, I have not tested this technique across two different platforms. If you have both a Mac and a Windows machine why not give it a try and report back in the comments if you found any adverse effects trying to use the same catalog on both platforms.
This is by no means an answer to the problem of a multiuser studio that needs access to the same Lightroom catalog. However, I find it to be a great solution when I need to bounce between my desktop and my laptop and want full access to my images. So, give it a try! If nothing else, you'll wind up with a free dropbox account that can be used for many things!