I would be immediately suspicious of someone asking large amounts of money for a rehomed dog. By all means they should be asking for a sum large enough to put off time wasters and people just wanting a cheap dog but £650 for a pug (as in one of these adverts) would have me immediately on alert.
Someone who didn't ask me as many questions as I asked them would also put me off. If they truly love the dog they are rehoming then they will want to know ALL about the home s/he is going to.
How they interact with the dog. A dog that has been loved is obvious and will enjoy interacting with their owner. An owner that loves their dog is more likely to be totally honest about them to ensure they go to the right home.
On that, someone that allowed me plenty of time to visit and see the dog before making a decision and didn't pressure me into taking the dog through emotional blackmail or claiming other people were interested in the dog. Anyone who did these things would send me running for the hills. You should be allowed to really see, walk, get to know the dog before taking him/her home.
I would also expect any owner that loves their dog to offer to take it back, no questions, if the rehome doesn't work out. One that washes their hands of the dog is not an owner I would want to deal with.
Someone for whom timing was critical (i.e. wanted the dog gone asap) would make me wary. Again, an owner that is caring will want to take the time to find the right new home.
Anyone who said the dog was perfect. NO dog is perfect and I could list, right now, 5 annoying habits about every dog I have ever owned, e.g. barks at the door bell, chews shoes, can be pushy when asking another dog to play, steals food off the counter, sneaks onto the sofa, has stinky breath, hates having nails clipped, is sick if eats too fast. I'm not describing a specific dog there (!!) just giving examples of where even great dogs have their quirks. An honest owner will let you know what they are.