Some medical schools run partnership schemes with local school, often those which are underperforming nationally. Newcastle and Manchester are examples of universities which run these sorts of schemes I believe but it’s not exactly that they have a preference, just that a certain number of places are sometimes reserved on these schemes and/or that the university may make contextual offers. People from these schools still have to meet the conditions of their offers, though. It’s not really a case of local student = more likely to get an offer as part of the general pool of applicants. If you don’t attend one of their partnership schools or meet the contextual conditions for an offer under one of these schemes, you’ll be treated the same as any other applicant, no matter if you’re local.
Anecdotally, it was rumoured at my school that if you applied to the local university you were likely to get a rejection just because of the school’s local reputation! Pure anecdote, though.
The one thing that may work in your favour though is if you’re a local student applying to your local university, you can probably give a very convincing answer at interview to “Why Sheffield/Keele/Leeds/Manchester/Newcastle?” etc! And, practically, it might be better for medical students to live at home to save money and some students have no choice in that regard.