I just dealt with this on the cat forum. There are some cat/dog/human transferable illnesses. Rabies and Leptospirosis, as are (some) parasites like ticks, mites, lice, tapeworms, hookworms, and heartworms.
dogs can pick up a parasite from anywhere Fox's have been and also slugs and snails. So if they go out in the garden they could catch it, its called french heart worm and its a parasite that gets in to the dogs heart and causes loads of problems including diarrhoea and foaming at the mouth. my dogs had it but i caught it early so i was lucky. my dogs are now on a spot on called advocate which is brilliant as it spots most parasites. Get them check for it its better to be safe than sorry.
Find a better vet. Get those pups to the vet and keep them there until they can make a correct diagnosis.
I have taken my dog to the vets everyday for the past week, all together within in 3 days they have subscribed her with an inflammatory injection, antibiotics, and one to stop the diorreah. However prior to this she had been having hormone injections, the vet put the sickness down to this, however I have 6 Yorkshire Terriers all together and the same symptoms has been passed onto two more of my Yorkies; worse than before. We have told the friend to move out now as he was causing trouble, so maybe I will see a difference. I had the idea of my dogs catching a bug off of the cats as they have not been out anywhere other than the garden and around the street; their normal walk, this is what I couldn
It's very unlikely that they have gotten anything from a cat. I would definately take them to a vet to get checked out. Did they possibly eat something different? Another thought ... you mentioned you had someone staying with you. Yorkies can tend to be pretty nervous dogs. It's possible the diarrahea and foaming at the mouth are being caused by the stress from the change in their enviroment. Take them to the vet to rule out all the options. Good luck!
I am not familar with any cases where dogs can get diseases from people who have been in contact with someones filthy cat.
The foaming at the month and diarrhea is a what is concerning to me. PLEASE talk you dogs to the vet. Good Luck!