The FAA experience requirements for the Private ticket include at least 3 hours of dual night VFR including one cross country of at least 100 miles and at least 10 night T/Os and landings to a full stop. After that, night VFR is just one of the privileges of the license.
There's also a currency requirement: to carry passengers at night, you have to have made at least 3 night landings in the last 90 days.
That said, it's not unusual for a rental operator or flying club to require an instrument rating for intentional night flight.