beszélgethettek
beszélgethettek is a Hungarian verb form. It is the past tense, plural, conditional mood of the verb beszélgetni, which means "to chat" or "to converse". Therefore, beszélgethettek translates to "they could have chatted" or "they might have conversed".
The conditional mood in Hungarian, indicated by the suffix -het- or -hat-, expresses possibility, capability, or
So, to break it down: "beszélget" (stem of "to chat") + "-het-" (conditional) + "-t-" (past tense) + "-tek"