True separation anxiety is fairly rare -- which is good, as it can be a very difficult problem to treat and usually requires a professional trainer's services, sometimes coupled with pharmacological help. Mild separation distress, however, is quite common among rescue dogs, and so there is a better-than-even chance that you'll have to deal with this at some point while fostering.
Nationwide, shelter dogs have a statistically higher incidence of separation anxiety and distress than the overall dog population does. Some of them may have been abandoned by their previous owners because of that behavior. Others may have developed it as a result of the instability in their lives. Every one of my foster dogs has shown some degree of (very mild) separation distress soon after arrival. Fortunately, every one of them was able to resolve this issue within about three weeks to three months.
It's not hard to see why a foster pup might feel frightened and anxious about your departures: you represent the primary source of security that this dog has known in her life. She has lost every home she had before coming to yours. Some of the people in her life had good reasons for sending her away (such as the transporters who brought her North and the caretakers who saw her through her pre-transport quarantine period), but the dog doesn't know that; she only knows that these people were kind to her and now they are gone. So she really, really does not want you to go away too.
(Incidentally, this is one of the reasons it's so important to make a good match between dogs and adopters. Every time an adoption fails and the dog bounces back to the rescue group -- or, worse, gets dumped at a shelter -- she suffers another bout of homelessness, and the upheaval makes matters worse. A temperamentally sound dog may develop separation anxiety. A dog who's already anxious may suffer worse symptoms. If your foster pup has any tendencies toward separation anxiety, be doubly careful in trying to place her with the right home.)
Sometimes what's diagnosed as "separation anxiety" really isn't. Dogs who have accidents in the house while you're gone, but also have accidents when you're home, are probably just not potty trained yet. Even if they only make mistakes while you're away, it's possible that the accidents occurred because the pup just couldn't hold it that long, and no one was home to notice that the dog needed to go out. Dogs who are generally destructive may just be bored, deprived of appropriate outlets for chewing, or inadequately managed. Dogs who bark may be frustrated, bored, or triggered by seeing "intruders" pass by the windows.
But if the dog's behavior is clearly linked to your departure -- not just a matter of trouble he might get into while you're gone, but signaled by stress-related behaviors such as panting, drooling, pacing, scratching at the door, whining, or distress barking as you leave -- then you are probably dealing with genuine separation anxiety or distress.
There are several types of separation anxiety. Pat Miller has separated them into isolation (dog is distraught about being left alone, but can be soothed by the presence of any other friendly person or even a companion animal) vs. separation (dog is distraught about being separated from one particular person and is not consoled by anyone else's presence) and differentiates "distress" (less severe) from "anxiety" (more severe). If your foster pup suffers from isolation distress, it may be sufficient to have another dog around in the house to help her feel safe. But if she has full-blown separation anxiety, you have a harder road ahead.
Mild separation distress, which is what most foster dogs will manifest, consists of distress vocalizations (whining, barking, yowling) and low-level stress signals (pacing, panting, yawning, lip licking, intense sniffing of uninteresting objects or places, scratching when the dog probably isn't itchy, and so forth). It can be successfully addressed by general relationship- and confidence-building coupled with "tincture of time." As the dog becomes more comfortable in your home and more secure in the knowledge that the daily routines include you returning predictably every day, her distress vocalizations and panting/drooling should taper off.
In the meantime, don't encourage her behavior: don't respond to the distress cries (as hard as this may be), don't make a big deal about arrivals or departures, and work on improving your foster pup's calmness and confidence with positive training exercises. You can help matters along by practicing some of the techniques described for moderate distress below. Additionally, getting her used to early, frequent, and short absences can help her learn that your being away is not the end of the world.
Never punish her for having accidents or chewing things while you're gone; by the time you get back and see the damage, it's too late to teach the dog anything except that your arrival can be a cause for anxiety. That message can worsen a mild case of distress, and even if it doesn't, it's counterproductive to your relationship with the dog.
Moderate separation distress can be remedied by a program of desensitization and counterconditioning. Essentially, you want to get the dog used to your comings and goings bit by bit, starting with the very first departure cue that puts him on edge, and teach him to associate those Not-So-Bad Things with Really Great Treats until he's not only accepting of them, but actually sort of looking forward to them. Then move on to the Kinda-Bad Things, then the Really Bad Things, working patiently to change his negative emotional associations via lots of awesome treats and games.
So you'd pick up your keys, give him a treat, put down your keys. Put on your shoes, play a few games of fetch, take off your shoes. And so on, gradually building up to brief periods and then longer periods of actually leaving his sight (and then the house). This will most likely take several weeks or months to resolve, and will require considerable patience and consistency, but it can be done. Patricia McConnell's booklet I'll Be Home Soon is a good resource, as is Nicole Wilde's Don't Leave Me! Randy Grim's book Don't Dump the Dog has a good (and very funny) outline for treatment as well.
Dr. Karen Overall's Protocol for Relaxation (free MP3 files for download) is another helpful exercise to practice with an anxious dog. It does not directly address the fears that underlie your foster pup's behaviors, but it does encourage calmness and a steadier mental state, which can help the rest of the treatment take effect more quickly.
Severe separation anxiety can cause tremendous property damage and physical harm to the dog as she desperately tries to escape confinement and find you again. This is the level where dogs hurl themselves through plate-glass windows, tear apart wooden doors, and break their teeth on the bars of their crates. It is absolutely a situation that warrants consultation with a behaviorist, a good professional trainer, and probably a veterinarian who can prescribe anti-anxiety medications. With any luck, you'll never have to deal with it... but if you do, bring in the big guns fast. This is a very difficult behavioral issue to resolve, and you'll want to have the best help possible if you decide to try tackling it.