Having grown up fairly close the first major (relative term, of course . . . it's mainly just a few closely grouped families) church of this kind, I can tell you a bit about their methods. I'm sure Ossai had a good view, but didn't see a lot of the prep work
First off, the poisons they drink are either generally very dilute, or not substances very toxic to people in small quantities. Strychnine is often cited as a drink of choice, and here's a link concerning toxicty . . . note the bottom of the page, in partcular.
Link. Great dilution plus some good prep renders this almost harmless.
As for the snakes, well, snake handling is an old art. Basic herpetology gives you some decent instruction on the matter. I knew students that safely handled copperheads and the like after just a week of classes. Also, the snake storage boxes generally aren't kept all that warm. Lethargic reptiles = more safety. Also, though I haven't witnessed it, I have heard that the snakes being handled are occasionally milked prior to demonstations, thus reducing the amount of venom available.