Oxford MFL tutor here. In fact we have very few native speakers enrolled in our undergraduate courses.
We are very very keen to have more students from state schools. Linguistic ability is only one component of the course, however. The MFL courses at Oxford (and Cambridge too) are heavily literature-based. Doing an MFL degree is not unlike doing an English degree, only with the added language component. So one thing we are looking for is students who in addition to being gifted at languages, genuinely enjoy reading and writing about literature. So an applicant may have stellar language skills, but be more interested in MFL and say, business studies. In that case they would find MFL courses at other universities that would probably suit them far better than the Oxbridge MFL courses.
Candidates' exposure to reading literature in the foreign language before they apply to Oxbridge will differ dramatically according to their school curriculum, and that's OK. Often pupils from state schools will have read less literature in the language than pupils from independent schools. But if they enjoy reading English literature for pleasure, then that's a good sign. I don't mean that the course is entirely literature-focused (the structure is half language, half literature, and there are also loads of opportunities to work on history, culture, philosophy, film, gender studies, etc). But the literary aptitude is important!
We don't expect people's linguistic skills to be anywhere near native-quality when they start the course. However, there is a language test to sit as part of the admissions process (these tests are now administered in schools), and good knowledge of basic grammar is essential. There are sample language tests on the university websites.
I just wanted to say all this as it would be very disheartening if state school students decided not to apply to Oxbridge because they had not had holidays abroad, immersion in a foreign country, etc! These things are nice but definitely not necessary. What we want to find are people with passion to study the language (and with strong potential of course!).