SQLite databases are just flat-files. What this means in layman's terms:
- I download your database as a file.
- I can modify it as I please.
- I'm not forced to synchronize to the newer version of the flat-file.
You have a choice: You either write your code to force periodic synchronization of the SQLite flat-file to your clients, or you use a dedicated DBMS on your server, such as MySQL, and force your clients to connect to that.