As you may know the supported OS is the Yocto one and the Shield Test Report was done using the Arduino IDE. Unfortunately there are none official tests with Debian nor using Linux.
In order to use a Shield and interact with Linux it will depend on the shield and the interfaces to be used (I2C, SPI, UART, PWM, analog or digital signals..). There are a lot of threads using these interfaces with Linux so I think it will depend on which kind of shield you are going to use.
Shields don't show up as devices on the Galileo. Only pins and IO ports do. Devices attached to USB and PCIe still behave the same they would on a PC though. Galileo is kind of an odd mix between worlds. There is the whole Arduino compatibility world and the Linux environment. Shield support is only exposed through the Arduino compatibility libraries. This doesn't mean you couldn't write your own drivers for them, but you don't get any help from the Intel software for that. It is likely that you would get better performance by writing a kernel module vs user-mode code anyway. Shields also don't have any kind of PnP feature and have the whole 5V vs 3V logic compatibility mess due to their origin being with Arduino.
My recommendation for things like WiFi and GPS is to use the USB and PCIe connectors as much as possible. Arduino doesn't have those so the shields often have to do things the hard way because of it.