The subject of exotic animals kept as pets seems to be an increasingly global problem. And though it was correctly remarked before that especially western countries are trying to restrict the problem by passing more and more laws, many people nevertheless can get the animals (more or less illegally) they want-if they have the money. A good example was a report I recently read about rich Saudi Arabians easily getting wild-caught dibatags or lesser kudus for their private zoos from Somalia; another is that of a Czech parrot breeder smuggling raven cockatoo eggs from Australia to the EU in his underwear or those Europeans keeping Chinese alligators and tomistomas as pets.
CITES and newer regulations make the animal traffic today harder than it was 20 or 30 years ago (when You could walk the street with Your pet cheetah...), but still the private sector of exotic pets is booming; look at American HPs like exotichobbyist or go to American animal or especially European reptile auctions; especially in the reptile/fish/arthropods sector there is nothing what You can not get if You have the right amount of cash. Even within states, official ruling on exotic can differ: while Kansas and Florida (the latter due to the many neozoas brought in over the years) are rather strict when it comes to exotics, e.g. Minnesota allows You to keep pygmy hippos and giraffes as "pets". Same is true for the EU; while the German federal state Hessen thinks of abolishing all exotic pets altogether (what rises the question about the definition of exotic pets, as neither budgie, tabby nor goldfish are actually native to Europe), a Belgium-based animal dealer shows on his HP that he can get You everything You want, may it be a Striped Hyena or a Marabou stork.
Although it seems to be ironic that the public eye watches every misdoing of zoos while millions of pets are mistreated and (more or less quickly) killed privately due to the dangerous combination of ignorance and incompetence (just think of the fish sold at the pet shops together(!) with the aquarium every day or the many, many rachitic reptiles brought to the vets...), one should not forget that in some parts zoos have played (and still play) a rather shady role in that exotic pet traffic; think of the destiny of some zoo surplus livestock, as mentioned in the book "Animal Underworld".
In the case of certain, possibly dangerous animals like big cats, venomous animals or large crocodilians and snakes, some states require future owners to aquire a
proove of competence - though this doesn't necessarily mean that "black sheep" are not around, keeping these animals without a permit/license.
On the other hand, it is also frustrating for responsible owners to be molested and harassed by a bureaucracy gone mad. Best example mentioned Hessen; it was now found out that the statistics displaying the numbers of household accidents involving exotic venomous pets that were used as justification for passing the law prohibiting the husbandry of possibly dangerous animals (more to follow...) were rigged up by an organisation called "ProWildlife".
I'm not saying that every animal species is suitable to be kept as a pet (absolutely NO!); I just want to point out that there are quite a lot of responsible and competent private owners around who can help more in establishing and maintaining ex-situ breeding populations of not-so-popular species like certain invertebrates, fish, amphibians/reptiles etc. than most zoos that have to focus on the animals interesting for the public. It's no wonder that f.e. the first successful rearing of many reptile or fish species (or cheetah; think of Rome 1965) has been accomplished by private owners, not zoos, who can dedicate more time and energy into those "not spectacular" species; not to mention important observations about feeding/behaviour etc. Better & easier interaction between zoos and these dedicated private owners could prove to be procreative; think of Münster Zoo's Turtle/Tortoise Project.
My advice would be a combination of carrot and stick: more severe and financially painful punishment of irresponsible owners and the illegal animal trade; use these fines to establish and maintain animal shelters for unwanted/confiscated former exotic pets. Make it more difficult for ignorant owners (and pet shops) to get animals; employ better educated staff at the customs authority, and improve the traceability of animal trading. But also make it easier for responsible and competent owners to maintain their breeding groups and work together with them to find the "black sheep" among them.
About the animals that should not be kept as pets: it depends on the individual situation; if a rich guy wants to keep lions, zebras or elephants and can provide excellent care, including a competent staff and can make sure that the animals are properly confined I wouldn't mind. Primates are a little bit different, especially in the case of the Great Apes, as mentioned humanisation could run contrary to the husbandry the animals really need. Not to mention the zoonosis problem...The question with all those private zoos is: what happens to these animals once the owner dies or can't afford the husbandry costs any longer-see Michael Jackson's Neverland Zoo...
And I can't think of a single sensible reason why a private person should keep an animal like a Black Mamba, an Eastern brown Snake or a Sea Krait...