Word for people who easily believe in rumors, and for a person who spreads them
Rumour-monger, gullible, and credulous are all possibilities:
rumour-monger - noun [ C ]
a person who spreads rumours
gullible - adjective
easily deceived or tricked, and too willing to believe everything that other people saycredulous - adjective [formal]
too willing to believe what you are told and so easily deceived
Cambridge Dictionary
I believe you are describing a gossiper.
According to Vocabulary.com:
"A gossiper is someone who talks eagerly and casually about other people. If you like to spread rumors and hear the latest news about your friends, you might be a gossiper. When you gossip, you talk enthusiastically about other people's news or business. To do this regularly is to be a gossiper."