I often listen to and pass on gossip about public figures and neighbours, telling myself it is just harmless talk. Is this the same sin as backbiting a friend?

You call it harmless talk, and that is what most of us call it. Let us look at it plainly.

Allah gives us a discipline for news that reaches the ear: a pause, before we believe it and before we pass it on.

“If a corrupt person brings you news, verify it, lest you harm a people out of ignorance and then become regretful over what you did.”

Qur’an, Al-Hujurat 49:6 · meaning

Notice what the verse is guarding. Not your standing, but a people who may be harmed while you were only repeating what you heard, and a regret that arrives afterwards, when the harm is already done. Elsewhere Allah describes the man whose habit is to carry words about, and tells us not to walk behind him.

“And do not obey every worthless habitual swearer, a defamer, going about with malicious gossip.”

Qur’an, Al-Qalam 68:10-11 · meaning

Now to your actual question, whether talk about a public figure or a neighbour is the same as talk about a friend. Allah settles it in the same surah as the first verse, a few verses on, and He draws no line between the near and the far. To mention an absent person in a way he would dislike is ghibah, wherever he happens to be.

“And do not spy on one another, nor let some of you backbite others. Would one of you like to eat the flesh of his dead brother? You would detest it.”

Qur’an, Al-Hujurat 49:12 · meaning

So the answer is yes. Idle talk about someone who is absent is ghibah, whether he is a famous name, the neighbour across the wall, or the friend you love. And when such talk is not only heard but carried onward from one person to the next, a second thing joins it, namimah, tale-carrying, about which the Prophet, peace be upon him, spoke plainly.

“The one who spreads namimah, tale-carrying, will not enter Paradise.”

The Prophet (peace be upon him), Sahih al-Bukhari and Sahih Muslim (Bukhari 6056 / Muslim 105) · meaning

Take that as a warning, not a verdict on you. This is a custom of the air we all breathe, and what we picked up from the air can be set down again. And there is something worth noticing in ourselves: the easy talk we allow about strangers slowly wears down our care, until the same ease reaches the people closest to us. Guarding the far is how the near is protected.

There is a quiet dignity in being the one in whose company reputations are safe, the place where a rumour arrives and goes no further. Your own heart is lighter for it.

One small step: The next time news about someone reaches you, let it stop with you. Say nothing of it, and turn the talk to something good. Notice how clean that silence is.