Even if you managed - somehow - to get DropBox to synchronize these files it won't work and most likely would just corrupt your database.
To make matters worse - let's assume you shut down SQL Server on the first computer - synchronize the files and then start up SQL Server on the second computer. If that worked and the second computer mounted the database - SQL Server would then upgrade the files to 2016 format and you could never go back to the 2014 version.
If you make sure you utilize the exact same version of SQL Server on both systems - made sure you stopped SQL Server before synchronizing files - then it might work and you might be able to get something working. However it is prone to a lot of issues and if you corrupt your database without having a good backup - everything you have will be lost.
The better solution is to create a backup file - place that file in DropBox - and use that file to restore on the other machine. It requires that you have the same version of SQL Server on both systems - but can be automated through a script that you run anytime you want to create that backup file.