You do not need server though FileMaker Server adds options not possible with just FileMaker Pro.
You can use the FileMaker Pro application to host your database from a computer and then FM GO devices can open it as clients of that hosted database if they can connect to the hosted database via WiFi or 3G/4G. The exact settings can be simple if they need to connect over the local network via WiFi. Connections via cell communication (3G/4G or WiFi from a remote location) require setting up port forwarding to enable access to your computer from outside the local network.
WIth FileMaker Pro as the hosting application, you are limited to a total of 9 clients (either FMGO devices and/or other computers using FileMaker Pro) at one time. Hosting with Server permits access by many more clients and offers many more tools/options for managing a hosted database that are not available with FM Pro.
You can also physically copy the file from your computer to a single phone. This can be done with the iTunes application or you can use certain other apps such as Drop Box.
You can host a database with filemaker pro or with filemaker server. You can also transfer the database to the device so it will run from the device. Itunes is the best way to transfer databases to Filemaker Go on your device. You can also email your database, or use such services as dropbox and or sugarsync.