In general, there is no way to determine a single "correct pronunciation" for words with multiple pronunciations. That said, as far as I can tell there is no basis for saying that Americans in general put primary stress on the second syllable of idempotent. If you look it up using OneLook Dictionary Search, you'll see that a number of other pronunciations are listed in American dictionaries. It's certainly plausible that some Americans say it this way, but I can't even find any evidence that this is an "accepted" pronunciation for most Americans, let alone "the" American pronunciation.
I found the following pronunciations listed in the major online dictionaries indexed by OneLook (I've re-formatted and standardized the transcriptions, using 3 different systems):
IPA: /ˈaɪdəmˌpoʊtənt/, American Dictionary: /ˈīdəmˌpōtənt/, Re-spelling: EYE-dm-POE-tnt
only pron in Merriam-Webster, 1st pron in Collins and possibly Dictionary.com (it doesn't distinguish primary and secondary stress)
IPA: /ˈɪdəmˌpoʊtənt/, American Dictionary: /ˈĭdəmˌpōtənt/, Re-spelling: ID-dm-POE-tnt
2nd pron in Collins and Dictionary.com
IPA: /ˈiːdɛmˌpoʊtənt/, American Dictionary: /ˈēdĕmˌpōtənt/, Re-spelling: EE-dem-POE-tnt
1st pron in Oxford Dictionaries, US
IPA: /ˌaɪdɛmˈpoʊtənt/, American Dictionary: /ˌīdĕmˈpōtənt/, Re-spelling: EYE-dem-POE-tnt
2nd pron in Oxford Dictionaries, US and Oxford Dictionaries, British and World English
IPA: /aɪˈdɛmpətənt/, American Dictionary: /ˌīˈdĕmpətənt/, Re-spelling: eye-DEM-puh-tnt
1st pron in Oxford Dictionaries, British and World English
Why it (often) isn't stressed like "omnipotent"
The word omnipotent comes from a pre-existing Latin adjective (omnipotēns, genitive omnipotentis). This became French omnipotent, which is the immediate source of the English word. English speakers generally perceive French words as having their stress on the last syllable, but this is a pretty unnatural position for the stress of English adjectives. Many adjectives taken from French have shifted the accent two syllables back, resulting in the stress on the third-from-last ("antepenultimate") syllable seen in the English adjective omnipotent. (This description of the stress shift is an oversimplification, and the details are complicated and not entirely clear; see the discussion of "countertonic stress" in this article by Piotr Gasiorowski: "Words in -ate and the history of English stress").
The word idempotent is a compound word formed in English from the Latin word idem and the adjective potent. In general, English compound words retain some degree of stress in the original positions on both parts.
A possibly analoguous example is the word polyvalent/-ce, another compound formed in modern times from classical components: it's usually pronounced "POLy-VALEnt", with the primary stress in the same place as the word valent/-ce and secondary stress on the first syllable of the prefix, whereas the word equivalent, which can be traced back to Latin aequivalens, has the stress on the third-to-last syllable ("eQUIValent").
Why the initial vowel has various pronunciations
There are different traditions for pronouncing Latin words in English. The oldest still in use is described by this Wikipedia article: "Traditional English pronunciation of Latin." In this tradition, vowel length in Latin has nothing to do with English pronunciation. Instead, vowel letters are pronounced as "long" or "short" depending on the surrounding letters and the position of the stress. In words like item and idem where the i is stressed and followed by a single consonant and then a single vowel letter, the English "long i" (IPA /aɪ/) is used. Traditional pronunciations are entrenched for many common words taken from Latin.
The pronunciation with a "short i" /ɪ/ based on the Latin quality belongs to a newer method of pronouncing Latin words in English, the "Reformed Pronunciation of Latin." This pronunciation was designed to be closer to the sound of Classical Latin. (It is not the same as classical pronunciation, because Classical Latin had sounds that don't exist in modern English. For example, the classical monophthong /eː/ is approximated in reformed pronunciation by the English diphthong /eɪ/).
The pronunciation with a "long e" sound /iː/ seems to come from the Italianate or "Ecclesiastical" tradition of pronouncing Latin (Italianate pronunciations are often used in singing, but rarely in ordinary speech).