17 feet would be ultra-safe. I do 14 x 14 on most of my trees and have very little problem with shading unless I don't prune. A keitt can fit very comfortably in a 17 x 17, and a lychee can easily fit into 17 x 17 with pruning. With zero pruning, most trees can fit into 25 x 25.
I think any mango tree can be kept productive at 14 x 14. If you're anti-nitrogen like Dr Campbell, then you can pull off high production at 10 x 10 :-).
Mango trees only grow from end of harvest to roughly end of Nov. With approximately one flush every month, you're looking at 2 - 4 flushes per year, depending on nitrogen levels, for productive trees -- which translates roughly to 2 - 4 feet of growth per year. So the annual pruning takes the tree from 14 feet to 10 feet in July, and the tree is back at 14 feet in December and ready to bloom.
The other thing to keep in mind when planning spacing is that mango trees only produce on the margins of the canopy. So, any space under the canopy edge is wasted both in terms of photosynthesis (it's all shaded) and fruiting.
Make sure to tour some of the orchards down this way that use tight spacing: Truly Tropical, Lara Farms, Fairchild Farm, etc.