I'm pretty sure it's all flavor and thus up to you to decide.
Officially in 4e, according to "Open Grave," the person turned into a zombie or wight isn't actually that person anymore. Their soul (and memories and personality_ goes to the shadowfell while the body reanimates. For totally mindless undead like skeletons and zombies, it is just a body walking around.
In the case of (most) intelligent undead, the body still has memories that it can use to guide it, but it is basically a new soul-less being. Think of it like a warped and corrupted clone; it has your memories but it has been corrupted by whatever disease/magic created it so the personality it manifests is going to be a twisted reflection of yours. Since it doesn't have a soul, it has a horrible torturous emptiness that it strives to fill by consuming flesh or draining energy or whatever. In 4e flavor text, these creatures are basically totally evil (though they may feel compelled to occasionally grant mercy based on the memories that torment them), but I like the idea of the newly formed being (saddled with memories that aren't really his) trying to "live" without resorting to evil. Maybe it could gain a soul one day through its studies or good works or service and finally be at peace. Then death wouldn't mean the end of its existence (i.e. total annihilation) but rather a release into a (hopefully) better afterlife.
Some intelligent undead like liches and vampires and vengeful spirits/walking dead actually do have souls and are the original person stuck in a dead body or in spirit form. In most cases this is done on purpose to gain power, right a wrong, stave off death until the being can redeem its immortal soul (to avoid going to Hell for example), or just live forever (and maybe avoid Hell without being nice).
You probably need to kill the undead being to recover the body so that the original soul can return and resume life. This could be morally problematic if the new being (an unfortunate ghoul for example) is not depraved villain but rather a lost and scared new life form trying to make sense of its surroundings (bwaahaahaa, evil DM plot point!). Alternatively, you could have the raise dead ritual create a new body for the soul of the departed. Then you might have an evil undead twin (hopefully it has a goatee) running around. Or maybe a pathetic twin that you feel sorry for. Lots of possibilities!
I can see how in some settings you might want to have all undead have souls (e.g. so that zombies are souls trapped in rotting bodies until destroyed and ghouls are tormented beings that now have to deal with a terrible hunger). That is totally up to you.