Thing is, it's hard to consider "locals" as one entity, would it be only due to inner French migrations in these regions since centuries and outer migrations since the XIXth.
Basque Country (Vasconia is more of an historical term, a bit like calling English Midlands "Mercia") and Corsica are the main preservers of a regional identity that include language. Basque language use remains essentially limited to both more aged populations and to school for the younger ones, with less than 1/4 in French Basque Country overall (much less between 25/49, which is roughly around 15%) able to understand it, which is (contrary to Spanish Basque Country) in constant decline.
Corsican language is in a better shape, with nearly 1/2 of the population claiming to understand it. But it's in similar decline nevertheless (UNESCO pointing that it's in danger of extinction).
Breton might be spoken by a bit more than 10 000 persons, understood by 300 000 persons.
Overall, while language plays an important regional identitarian role, they're not exactly thrilling and no one knows the demographical rise in use that Welsh does knows.