The idea of bilingualism seems mostly to fascinate those who are not bilingual. I have seen estimates that about half the world's population is bilingual, so for many it is pretty normal and I do not expect they think about it. In monolingual communities there are many who say they do not have a gift for languages; if they lived in bilingual communities they would have the gift!
The way people become bilingual includes:
A. Being brought up by parents who speak language A in a monolingual community that speaks language B.
B. Being brought up by parents who speak language A in a bilingual community that speaks languages A and B.
C. Being brought up by one parent who speaks language A and one who speaks language B in a monolingual community that speaks language A or B.
D. Being brought up by one parent who speaks language A and one who speaks language B in a bilingual community that speaks languages A and B.
E. Being brought up by parents who speak language A in a community that uses language B for education and official purposes.
F. Moving (especially at a young age) from monolingual community A to monolingual community B.
I suspect it is only those who come in group D where languages A and B have equal status whose language experience will be the same or nearly the same in both languages.
Children who come in group A may start to identify with language B and even speak it to their parents. I have personally witnessed on a number of occasions conversations between an immigrant parent and child were the parent speaks one language and the child another.
Those who come in group B, where although the community is bilingual there may be a preference for language B, may feel a resentment towards language B and avoid speaking it whenever they can.
Asking bilinguals to describe what it is like to be bilingual is a bit tricky. When they start to think about it they are not sure. There is the oddity that you may feel you are a (slightly) different person when you change languages. I am by no means fluent in Spanish, but when I speak it I feel the duende coming on!