Hello Joe,
I have only 7 months experience now with humanure composting. I'm in a temperate
climate with 40+ inches of rain yearly. (so no tropical experience)
I don't have my pile covered and it doesn't seem to be an issue. A 1" - 2" rain will cool the pile but it comes roaring back. I don't see evidence of leachate escapinng. When I started the pile I built a dirt/clay bowl on the bottom, then added leaves/grass ~ 18" as a biologic sponge. I have more issues with the compost getting dry and thats with daily additions of food scraps and rinsing containers.
Joe Jenkins, the writer of the Humanure handbook does have a roof over his piles. Have you seen his youtube videos, website, read the handbook?
I use a spray bottle to wet toilet paper at the
bucket toilet then scoop on biomass. I barely see evidence when I dump the fresh buckets. I also add on junk mail/paper towels that have been soaking in
water on top of the weekly emptied 5 gal buckets. They seem to be decomping. I am generous with the top cover (I don't want to see or smell paper, egg shells or veggie scraps.
I'll find out in 2 years when I open the pile to use it how the paper did. (my climate is a 2 year cycle- the tropics may be faster)
Good luck