We have all see on TV the judge instruct jurors that during trial they are not to speak about the case with anyone, even other jurors, unless all jurors are present and they are deliberating. However, contrary to the example given about England, in the U.S., those restrictions evaporate at the end of the trial. After a trial concludes, the court has no continuing control over the jurors and could not impose lasting restrictions without it.
The Constitution provides the guarantee of trial by a jury of ones peers. In the U.S. for all general civil cases and all criminal cases, we have public trials. (special courts and tribunals are created to deal with cases involving classified information and issues of national security, and the courts have mechanisms for handling trade secrets, etc. to insure that information is not presented to jurors) So in that sense, there is nothing a juror could be exposed to in during their service as jurors that would require any type of continuing restriction.