Microclimates are areas where the area climate is different from the larger, surrounding area. I once camped near a lake as a boy and was informed by a counselor that when weather reports indicate a chance of rain, they get rain virtually by default. The natural rise in humidity from the lake evaporating and collecting rainfall made the area more rainy.
In a similar sense, desert trees live off of water, whether from groundwater and oases, and oases naturally are wetter areas with more plant life, plants including trees. A lot of your creatures, though not all of them, will live in or around such relatively tree-heavy oases.
So in this case, the oasis would be a high-magic microclimate, with a higher magic level than the surrounding area. Even if the magic diffuses like oxygen through the atmosphere, being close to the source means exposure to more magic. But there are other examples of little microclimates, as desert snails demonstrate.
Option 2: Have the animals be part of the system.
If the trees produce magic, is that magic sequestered in parts of their body? Trees like other plants naturally take nutrients from the ground and create energy from sunlight in the form of sugar, and both sugar and nutrients can be found in their fruit, leaves, and so on.
And desert trees, like date palms and Joshua trees, do produce fruit. Fruit that will be eaten by native desert wildlife. It'd make sense if they incorporate those magical energies and ordinary nutrients into their bodies, because regular life does something similar through digestion. And maybe-just maybe-as animals feed on the tree's fruit or foliage, they become magical themselves and can help share the load of magic distribution.
I feel like that's not exactly what you're going for, but perhaps only desert creatures with a reliance on the tree for food and shelter, those especially tied to the magic trees, can become magic themselves? Or perhaps only certain desert animals in magic-poor environments have developed the capability to produce magic as trees do, a la leaf sheep, and are preyed upon by the others. Such animals would even help explain where the trees are getting magic, and would set up a magic cycle that-again-is not present outside deserts. Easy enough to explain when biological specialization-like that of the koala for forests-can make them unable to survive in different environments like the arid Outback.
Either way, I hope this helps you!