EU privacy laws prevent the routine sharing of information about criminal records (etc) other than to other EU countries that have signed up to relevant parts of the Schengen Agreement.
So while, say, France can currently routinely hand over information to the UK, it can't hand over such information to the USA. If we leave the EU (and thus our right to participate under the Schengen Agreement) we'll be in the same position as the USA now is. (i.e. France will only hand over information upon receipt of a specific request, about a specific individual, rather than routinely).