In general, EFL students are taught the two main ways of pronouncing the determiner "either" are the British [ˈaɪðə] and the American [ˈiːðər] varieties. However, I've repeatedly heard from specific North American individuals (face-to-face and on TV) another pronunciation: [ˈaɪðər]. My specific question here is: does that phenomenon take place because of some dialectical/idiolectical/sociolectical background? Thanks!
Many Americans do indeed use the latter two pronunciations interchangeably; one individual will use both, on different occasions. I don't agree that there is no difference at all, however. To me and I think many Americans, there is a very slight sense that "eye-ther" (American "long i") is more formal, elegant, higher-class than "ee-ther" (American "long e")—simply because it sounds more British to Americans.