Ciao Baby Bella !
The fact is that Spanish has more local variations and accents than English. With the letters "Z" and "LL" these differences are more apparent.
In Spain - and only in Spain - "la zeta" (letter Z) is pronnonced with the tongue between the teeth, sounding between "th" and "f"... so Spanish people say "Ibiza" quite like "Ibeefa" ou "Ibeetha".
All Latin American countries that speak spanish (no, Brazil is not one of them) pronounce la zeta without the tongue blocking the way, sounding like "s"... so the "international" sound of it is "Ibeessa".
The knowledge of Spanish that most North American people have are influenced by the presence of immigrants from Mexico, Central and South America, who all say "Ibeessa".
On the other hands, Brits and European in general are closer to speakers and teachers from Spain and tend to reproduce the strong native Spanish accent that goes like "Ibeefa".
So that´s the difference. They´re both right. It´s the same sort of difference that words like "better", "either" or "thirty" have in US/UK.
In my opinion if you don´t speak Spanish that well it will be much better and easier to say Ibeessa, because the Spanish way is not EXACTLY "Ibeetha" or "Ibeefa", and you´ll sound too foreign if you try.