Mange is VERY contagious. I am VERY surprised the Shelter did not know that. However, they want to get rid of the dog as soon as possible. It may spread to their other dogs. Our 2 separate shelters (Humane Society/City Shelter) BOTH had to shut down due to Kennel cough last year. However, they did allow my friend to adopt a dog a few days after the "quaranteen" and it had Kennel cough. That is contagious as well. I knew the second she adopted the dog it had it. Ironically, the policy at the shelter is that the dog must go back to their vet for nuetering/shots prior to full adoption. It still had not been treated for Kennel cough! A few days later when she went to pick the dog up, I mentioned it to the shelter. They claimed the dog had come in shortly after the quaranteen and hadn't shown any signs. Then the dog started coughing and they agreed that it needed possible treatment. We had to take it back to their vet and they charged my friend over $100 for medications.
The good news is that the dog got treated and was well within 2 weeks. The dog (Scruffy) was a gift to my friend's 3 year old daughter. The kid talks about Scruffy ALL the time and just loves the dog almost to death at times if your not watching!!! I've even gotten to babysit Scruffy a few times!!! He's a cut-tee!!!
I do encourage the adoption of this retriever. However, I do caution listening to the Not contagious to healthy dogs. We never know for sure if our dogs are healthy 100% of the time. My suggestion is that adopt the dog when the condition is finally under control. Buy throw away bedding or cheap bedding that won't hurt to toss out at the end of the month. WASH HANDS IMMEDIATELY AFTER PETTING THE DOG AND DO NOT PET THE OTHER DOGS UNTIL HANDS ARE WASHED. This is how the mange is spread into "healthy" households. I lived out in the country and we often adopted "mangy" mutts. However, we made sure the dog stayed outside until the condition healed. It could easily get into the carpet and spread. Gives baths atleast once a week. There should be a shampoo available for treatment. May ask the vet or get some kind of shampoo to treat the itching. Avoid FLEA and TICK shampoo's. It may hurt their delicate skin.
All in all. The dog should be able to adopt within a good 3 weeks to a month's worth of treatment. I think it is mostly cured within a month with proper treatment. I hope the dog finds it's way home where it is clearly wanted. Good luck!!!