Yes, Jupiter and Saturn have their own radiation belts. Jupiter's is extremely powerful, which is why all of Juno's sensitive electronics had to be radiation hardened and shielded inside a vault. It's also why Juno is going to be on a polar orbit doing flybys of Jupiter, instead of orbiting around it closely.
These magnetic bands are indeed separated into inner and outer portions. From Wikipedia:
Jupiter's magnetosphere is traditionally divided into three parts: the inner, middle and outer magnetosphere. The inner magnetosphere is located at distances closer than 10 RJ from the planet. The magnetic field within it remains approximately dipole, because contributions from the currents flowing in the magnetospheric equatorial plasma sheet are small. In the middle (between 10 and 40 RJ) and outer (further than 40 RJ) magnetospheres, the magnetic field is not a dipole, and is seriously disturbed by its interaction with the plasma sheet (see magnetodisk below).
Jupiter's inner magnetic belt was mapped by Galileo, showing it contains high-energy electrons. Eruptions from Io result in the middle portion of the magnetosphere containing huge amounts of ions of various elements.
Saturn's magnetosphere is likewise divided into separate parts. However, there are 4 regions to Saturn's magnetic field, and they are separated by the kinds of particles they contain. These particles result from interactions with Saturn's various moons, as well as its ring system.
The Galileo probe discovered that Europa does produce a weak magnetic field, as a result of its subsurface ocean. This ocean has an induced current created from Europa moving through Jupiter's magnetosphere, which in turn creates a magnetic field. This magnetic field is actually one of the things which caused many scientists to theorize a subsurface ocean.
However, these belts are not "Van Allen" belts. That name is reserved strictly for Earth's radiation belts.