This is the same as trying to share a database file between two users via Drop Box.
Your missing data or changes will be found in the second copy named with [conflict] as part of the file. Trying to share the database using this method will generate a large set of such copies over time and each one will store a different copy of your data--resulting in an unusable mass of inconsistent data.
You need to host the file from a copy of FileMaker Pro or server over your network and then use Open Remote to both access the same database file.
I also found that Google Drive was slow to sync (if it did). Switching to DropBox did alleviate that. But I am just storing my recipe database between two different Mac laptops I use. I would definitely follow Phil's advice on this.