Thank you for your post.
FileMaker Pro can share a database over the network that up to nine additional users can access. If the computer is outside the network temporarily, then it will lose connection and be dropped from the database. You won't need the server product unless you plan to use more than 10 users.
Another option is Instant Web Publishing. That allows up to five users on the internet to access the same FileMaker Pro file, as long as the host is running FileMaker Pro and has a static IP address. If you need more than 5 users (per 24 hour period), then you need to move to the server product.
Let me know if you need further clarification.
You may also want to consider using a hosting service which would allow you to have your databases hosted on FileMaker Server, on a server-class setup, without having to invest in such yourself. Hosting off FileMaker Server can be much better than hosting peer-to-peer with a copy of FileMaker Pro.
I have a hosting service, so contact me privately (top of this page - right side - X Messages) for more information.