I can think of a few possible words:

Eavesdropper (noun) -- someone who intentionally listens to a conversation that should be private between two or more other people.  An eavesdropper usually tries not to let the other people know that they are listening to that conversation.  (For example, in a restaurant you might eavesdrop on people who are talking to each other at a nearby table).

Nosy (adjective) -- a nosy person is too curious about other people's lives and what they are doing.    A nosy person may watch other people carefully (for example, a neighbor that looks out her window to watch what the neighbors outside are doing).  A nosy person might try to listen to other people's conversations when they walk nearby.  A nosy person might ask too many questions or ask questions that seem too personal (For example:  "So...the two of you got married last summer.  Any plans to have a baby???")

Busybody (noun) -- a busybody is a nosy person that listens or looks or has conversations to learn details about other people's lives...and then talks!  A busybody might share that information with a different person.  Or a busybody might give opinions to you about your life and what you should do, even though you haven't asked for their opinion and maybe don't want to hear their opinion.