The 'i' of side is not supposed to be pronounced differently from a standard Danish 'i'. The Danske Ordbog gives a pronounciation of [ˈsiːðə], which is the same 'i' that vi uses as well (just longer). It might be darkened a little by the following ð, but it's still far from the English "sir".
I can imagine a vowel hue change (is that the right term in English?) in front of 'r', because the Danish 'r' is a very vowelly sound, but the ð doesn't really do that. Do you have more examples?