I would urge caution on a private rehome, and insist on full access to veterinary records and a trial period, starting with simply taking the dog out for a walk. There are some issues that only become apparent after some time spent with a dog. I would also consider having the dog's temperament independently assessed by an APDT trainer.
I say this as someone who adopted a dog from a former flatmate - obviously a bit different as I'd already lived with DDog, but if the old owner had been giving a description of him he would have missed out a lot of important things behaviour wise as he was a bit inexperienced (frustrated greeter, reactive towards certain forms of traffic, mild resource guarding around dogs, no basic training, a seasoned bin raider...). He'd also been burying his head in the sand about veterinary things - no booster jabs, microchip out of date, what I thought was patella luxation ignored etc. He wasn't a bad person, just out of his depth and in a situation not of his making, but if you met DDog in the house you could believe he was the perfect dog like I did when I viewed the flat
I've also rehomed small furries from the small ads. I've had people not even ask my name let alone what I plan to do with it, different species handed over to the one in the picture and one that was being kept in a bucket and died in an expensive fashion three weeks later. I doubt the quality of people rehoming dogs in similar ways is much better.
The difference between the dogs from proper rescues that she rejected on the grounds of having too many issues and this one is that this dog is a total unknown - the proverbial pig in a poke.
If they didn't want to take it back after a month, it would be hard to force them, and you'd fear for what happens next to the dog - they may simply dump it, and it would probably end up in another unsuitable home. How much are they asking her to pay for the dog? I wouldn't agree to pay any more than I was willing to lose, and would plan to rehome via a proper rescue if she really can't cope.