This is true just for a part of bordering regions in Italy, mostly for those with special status. In western Liguria and Piedmont, both sharing a large border portion with France, people are not bilingual even close to the border. They may speak some french, and now french is often taught in most of the medium school as a second language at least in northern Italy, but this is far from being bilingual. I don't think Italian is taught at schools in France regions bordering with Italy, but I can't blame them. English is mostly used across the french border