Each tree gets its own compost bin and a veggie plot gets several. Good idea?

By filling a veggie plot with multiple compost bins, and each tree getting its own bin, the idea is to let the liquids & nutrients of the compost run directly into the vegetable patch and each tree. It’ll also provide constant moisture to the soil, which is important in drought territory like mine, and increase worm activity.

But are there problems too, like too much constant moisture rotting roots? Too much nitrogen from composting greens which prevents fruit growth?

Depends if you are aiming for, or care about, getting hot compost. Small scattered deposits will obviously break down, but in small volumes won't get hot and won't break down as fast. They also potentially attract more rodents, and over a wider area. Some people bury compostable material directly in the beds. You could potentially dig a trench, fill from one end covering as you go. Then plant on it.

hau Tim, I will go on the assumption that you are trying to "condition in place" the soil of your garden beds and trees, this is based on your description.

Some things to consider; a proper compost heap will not keep the soil beneath it constantly soaked, it will remain "dampish" but it won't or shouldn't rot roots of plants growing right next to the heap.
If you are having plant roots rot it is more likely from damping off (a bacterium) than from far to much moisture coming out of the compost heap.

Compost should feel moist but not wet or soggy if you dig into the heap and grab a handful, you should not be able to squeeze any moisture out of it, but your touch should feel damp.

The trench method brought up by Michael is a great way to incorporate small quantities of compostable materials but it isn't going to actually condition the soil the way building a heap on the surface will.

If you are looking to build heating compost piles, make one per bed instead of several, the smallest heap I've ever gotten to heat up properly was 3 ft. cubed, larger is better when it comes to compost heaps.
Another method that would work, but does require quite a lot of compostable materials is the windrow, this is a long heap at least 3 feet wide and at least 3 feet tall and extending however long you desire or can manage.
Windrows do have the disadvantage of needing to be moistened more frequently than a normal square type heap.

Redhawk