Hmm, not sure about most schools. I would guess whoever they think can gauge the levels best, or the class teacher at the beginning.
DH is French, so he sets all the work for the fluent/fluent-ish French speakers as he can guarantee a high enough level for them, and often brings back work from France to use with them, to supplement the GCSE/A level stuff. Obviously, French writing exercises for French children being more demanding than for those learning a second language in England.
The Spanish is taken care of by the Head of Department, who is a fluent Spanish speaker, but isn't Spanish, and I know she consults with other teachers (for example, their most junior language teacher has the best Spanish) to make sure that it's ok.
The bilingual students also often do a lot of work with the language assistants if the teacher is occupied with the rest of the class, as they are always native speakers, so can work on the higher level work with them.
To be fair, there's probably a much higher degree of independent learning for them, the teachers just don't have time. I know this year, for example, that none of the bilingual students will be able to work with actual MFL teachers during their timetabled lesson, as the MFL department is really stretched and they don't have the free time to do it. So they'll be in class, with hopefully the assistants at least, but even the assistants have to work with the non-bilingual children a fair amount.