Is one passive voice and the other reflexive?
To convey the meaning of English "they say", two constructs are used:
- Как говорится: с глаз долой ― из сердца вон! // "Out of sight, out of mind, as the saying goes"!
This (passive voice and reflexive) construct is used when citing an old proverb or saying.
- Говорят, если каждый день съедать по яблоку, будешь потом когда-нибудь очень здоров. [Юрий Коваль. Сиротская зима (1980-1993)] // "They say if you have an apple a day, you'll be very healthy, one day, someday".
This (indicative and impersonal) construct is used to convey hearsay evidentiality, that you are telling something you learned from the others rather than seeing or deducting yourself.
First of all, not "Говориться" but "Как говорится" (without soft sign) that is it's not an infinitive.
Second, "Они говорят" is a personal form in Russian. To indicate that "people often say smth." we say simply "Говорят..." (BTW. some indefinite pronouns can be used to achieve a desired level of impersonation as well, i.e. "Некоторые говорят" = Some people say, "Люди говорят" = People say etc. But "Они говорят" is always(!) personal).
So correct forms are "Как говорится" = It is said that... and "Говорят" - they say...