My aunt lives in the US and holds both UK and US citizenship... and she still has both her UK and US passports... so I don't think your work mate got the whole story; either that, or she misinterpreted the story.
If her aunt did have to 'give up' her passport, that would mean that she would also have had to give up her UK citizenship (which you can only do by filling out a form, paying a fee (currently £208) and officially renouncing it in front of a UK government official... it's not something you do lightly)... so it would now be impossible for her to be a UK citizen living in the US, because if she had given up her passport, she would no longer be a UK citizen, she would just be a US citizen.
Unfortunately, ksand24, for some reason, there are loads of misconceptions with citizenship and passports. I am eligible for dual citizenship from being born in the US and my parents' country of origin and understood since I was little that dual citizenship was allowed. However, so many people tried to tell me differently that I started to believe it
Even now, since I have two Americans, one expired with my visa and the other valid, I get officials asking me to check and see if that is 'kosher.'
US makes it quite hard to lose American citizenship
Funny thing, for example is that many dual Japanese-American citizens found a way to keep both even though Japan doesn't allow dual citizenship. Apparently, they inform the Japanese consulate that they are keeping their Japanese passport, but since the US doesn't regard 'force' as a valid means of renouncing citizenship, they get to keep their American citizenship as well.