Personally I've always said people when saying the plural of person but I here so much on the TV or even in person the use of persons.
It sounds gramatically incorrect to me but I don't know everything. So is one correct, or are they both right?
I think it depends on the context. Seems to me a number of specific individuals might be better referred to as persons, whilst a general group of folk is probably better referred to as people ?
It used to be that 'persons' was often used when speaking of a number of people who can be counted and 'people' when speaking of a large or uncounted number. For example, it was often said that a victim had been killed by 'a person or persons unknown'...ie clearly a horde of murderers was not involved!
Nowadays, 'people' is almost invariably used in everyday language, but 'persons' is certainly not "grammatically incorrect".