Well, it figures quasi-simplistically: Italki, as in “I talk” is herein compounded because it’s a web- addressed similarity reference to dot-com. However the consonant ending would make that other “i” sound differently, also “I” is a capital letter and “i” pronounced as “ee” ‘s ‘normal’. Which most English speakers would know if the capital letter of "i" is capital, they'd pronounce it differently than if it were 'normal'.

Also former nouns were of the same kind in pronunciation: “Walkie-talkie(s)”, which were well-known devices in all over the world, although not an official name. I think this is still remembered to those, thus making “Italki” an associative noun to the pronunciation thereof.

I'd say eye-TALK-ee to you, but ee-'talk-ee  to a Russian person. We are used both to classical Latin spelling and to English words.
This leads to confusion, the only unambiguous way to call letter 'i' being 'ee-with-dot'.

"Eye-talk-ee"

The founders probably pronounce it as /aɪˈtɔːkaɪ/    I pronounce it  /aiˈtɔːki/ and I agree with Jonathan's point about rhyming with "walkie-talkie." Independently of that (and the fact I love the way people from the Netherlands speak English), I have some arguments to support my position, but it's time for me to get back to planning my lessons.

I pronounce almost the same way as you, George. *I-Talky*