Yes and no.
There are two ways to share a filemaker database.
1) You can host it from one computer by opening it with filemaker pro or filemaker pro server. In that case each person needs a copy of filemaker pro to access the database. This approach is the easiest to set up and gives you the most control over the user interface.
2) You can use a copy of filemaker pro or filemaker pro server to publish the database to the web. In this case other users would access the database via their web browsers. There are two forms of web publishing: Custom Web Publishing (CWP) which requires filemaker server and an interface designed using PHP. Instant Web Publishing (IWP) attempts to directly reproduce existing filemaker layouts in the browser. It takes less effort to create but requires filemaker server advanced if you plan to have more than 5 users simultaneously accessing the database. It also is unable to reproduce all the interface features of using filemaker pro as described in option 1 above.
If you search this site, you can find out a great deal more about each option.
You may also want to consider using a professional FileMaker hosting provider, such as ourselves:
We provide FileMaker Server Advanced included in the price of our hosting packages, the server hardware, upgrades, monitoring, etc. so you don't have to. Feel free to contact me here or off-list to jmay(at)pointinspace.com with any questions you may have.