Two words: FileMaker Server.
Longer answer: Attempting to "share" FileMaker databases using any kind of file system / directory sharing, such as DropBox, Sharepoint, etc., is an invitation to corrupted databases. If you like your data ... don't. You can use a hosting service (there are several excellent companies out there) if you don't want to run the server yourself.
Thanks for your sound advice-- the fear of data loss or data corruption (apart from my own technical incomprehension) was one reason why I have not done this before. Maybe I'll go back to Plan A and the terrestrial rather than cloud solutions, or go Filemaker Server route. Thanks again.
FileMaker Server is designed specifically for the task you are trying to accomplish. If you aren't comfortable hosting the server yourself...use a hosting service. You can access your db from just about anywhere. There are some performance considerations depending on what you are doing with the solution...but for most users, little change needs to be made to work well.
Mike's caution has to do with other processes touching the open, live file while you are working on it. Can happen with sync software, anti-virus, backup software...etc.