Does she have separation anxiety in other situations? It sounds like stress to me, likely she doesn't have control over it happening.
I'd completely remove all high value things from their environment. No toys, no treats, chewies, etc. laying around for them to fight over until you teach them not to guard. Do not feed them near each other. Do not give them anything to guard from one another.
Basically you want to reward the dogs sharing space/stuff without fighting over it or guarding it from each other. Guarding is one behavior that is made worse with punishment. You take the stuff away and it reinforces to the dog that it needs to guard this cool thing because it keep losing it.
Does your dog allow you to take things? Or does it guard stuff from you too?
One thing that will be very useful is a solid "Drop it" command. You teach it by playing a trading game. Get some amazingly wonderfully awesome treat. Give the dog a chewie toy or something it likes. Tell the dog "Drop it", remove chewie toy from dog, slip it an amazing treat, then give the chewie toy back. This way the dog learns that if it gives up its cool things, it gets even better things, and then it gets the old cool thing back. Your dog will be happy to drop it whenever you ask. After it learns, you don't need to give the dropped thing back every time. It's useful to practice it once in a while to keep them fresh, but my dogs both will just drop it and not hold it against me if I keep it now.
After your dog is really solid on that, you can slowly introduce the other dog into the mix. First start out with it in the same room, but not interacting (have it on a leash or something so it can't get into the middle of things). The dog will learn that he doesn't need to be worried about the other dog's presence. You can also give him amazing treats the instant your other dog walks into the room. So he'll associate the presence of your other dog with getting cool stuff, rather than worrying he's going to lose his stuff.
The very last stage is a three-way trading game between the two dogs and you. You're the dispenser of amazing awesomeness. I trained this using a low value toy and hotdog bits at first, then worked up to a smoked pig ear and hotdog bits.
Basically, you hand one dog the toy. Then you give the other dog a bit of hotdog. Tell first dog to drop the toy, and trade him a bit of hotdog. Stick the toy in the second dog's mouth. Give first dog another bit of hotdog. Tell second dog to drop it and trade a bit of hotdog. etc. etc. etc. Mine got SUPER excited to play this game. Now they can eat out of each other's bowls at the same time and choose to gnaw the same bones at the same time without fighting. You DON'T want to rush that step however. It's the very last thing you do with them, and only do it when they are totally bombproof with the other foundation games.
It's going to take some management to keep them from practicing the guarding behavior, but if they don't get a chance to do it while you're reconditioning them their behavior will change must faster and be a lot more solid in the end.
