The two words are usually pronounced with "ye" : (i)eu , (i)este. It's a short "i" , just because it's easier to pronounce. The same happens with "el" (he) and "ea" (she) - pronounced (i)el and (i)a. ( "ea" is pronounced almost like "yeah" ) It is not a mistake to pronounce "e" like in egg , but it is a bit uncomfortable to pronounce and it sounds forced . However, i think that "eu, este, el, ea" are exceptions : most of the words that start with "e-" are pronounced with "e - egg" (element, electric , exemplu , exercițiu ....). So in short: pronounce "ye" for the short words , and "e - egg" for the long words.

It sounds like the Russian "е", so like the English "e" in "yes". It can vary in other words, like "element". People in Wallachia pronounce a clear "e", like the one in "egg", while those in Moldavia tend to use the "e" from "yes". But for "eu" and "este", I think everyone uses the "e" in "yes". Otherwise it would sound strange to me.

in words like eu or este , e sounds like the e from egg