Depending on the structure of your tables and number of users modifying the same set of records in different ways, that can be very, very messy.
It all depends on how you use the data and the structure of your database. If you don't want a brute force "latest update rules" type of synchronization, you often get a case where a human has to look at the data during the merge to decide how a given record wiith mutliple changes by different users should be merged.
If you can possibly host the database from a server with the users connecting to the hosted database, you avoid a possibly huge headache here.
The users will be located in different cities/ states. Could a server be used in that situation?
Sure thing! My company has been providing FileMaker Hosting services since 1999. Users can connect to your databases hosted on our servers from anywhere in the world, and edit data simultaneously. We provide the server hardware, FileMaker Server Advanced licenses, updates, monitoring, etc. so you don't have to. Please contact me privately if you want more information.