|MadSci Network: Microbiology|
The reason you can't find an answer is that this is not a "researchable" question. That is, most judgments about cleanliness in this situation can't be answered experimentally. We aren't really thinking about dirt, but about bacteria and other microorganisms, and even these aren't all bad.
Because most people don't have the means to work with microorganisms as I do, they judge mouth cleanliness by whiteness of teeth, and breath that doesn't smell bad. Dog and cat breath is seldom as nice as the breath of people who have just brushed their teeth, even though some owners do brush their dogs' and cats' teeth. Breath smells have something to do with what dogs, cats, and people eat — other than foods like garlic and onions, people's food probably causes less bad breath than raw meat, which is what dogs and cats evolved to eat. Of course, most of what owned (that is, not stray) dogs and cats eat these days is not raw meat, but my cat of many years ago really wasn't interested in anything else. Dogs and cats probably don't find people's breath pleasant, either. All the same, if your friend tells you that you have "dog breath," you know this is not meant as a compliment.
So, what about bacteria and other microorganisms? Saliva, and whatever other juices may be in the mouth, can contain and transmit bacteria, viruses, and parasites, in about that order of probability. However, the majority of bacteria that live in the mouth are harmless, except that some of them cause tooth decay. A few kinds of bacteria that are sometimes there can cause illness and may be transmitted by licking, kissing, etc. Even when we are sick (maybe with strep throat, which is really serious), these bacteria probably don't outnumber the harmless mouth bacteria, but it doesn't take a lot of them to infect someone else and transmit the disease. So, the concern is not about total numbers of bacteria, but whether any of them are harmful; the answer to this often changes from day to day.
Viruses and parasites tend to be especially species-specific. For example, most viruses that infect people won't infect dogs or cats, and most of their viruses won't infect people. There are a very few parasites of dogs, and especially cats, that can infect people, but these are generally not in the animal's mouth. Even many bacteria have strong preferences for host species. If your dog kisses you when you have strep throat, it is not likely to get strep, but your mom might get it from you by a kiss on the mouth.
So you see, the only way to look at the cleanliness question is, "What is the likelihood of passing a disease from a dog, cat, or another human to me?" Two hours after I've eaten, my mouth may be really loaded with bacteria; but these bacteria are just enjoying my dessert and can't harm anyone, including a dog or cat. On the other hand, if my cat had been out on patrol 2 hours earlier and had maybe sampled a sick bird or mouse that it caught, it might have Salmonella or something else dangerous that I could catch. Cats don't lick and kiss people as much as dogs do, and dogs are generally less interested in small, sick animals than cats are. Dogs and cats that don't get outdoors are less of a threat in this way, but they don't have as much fun, either. Finally, dogs and cats generally don't greet each other mouth-to-mouth as people often do, but I'm not sure their style is really any cleaner than ours.
Try the links in the MadSci Library for more information on Microbiology.