Am I supposed to care about friends of my sister?

Yes.

It seems kind of exhausting to be invested in strangers
you answered your question in your first post OP. It is a complete waste of your energy to invest in strangers. Focus that energy in your own life or at least people you know! Saying that I am here commenting on a mumsnet post written by a stranger! 😂
by and chance does your sister/ did your mum watch soaps? I ask because my mum is a drama llama and will get so caught up talking about the lives of others and being over dramatic and I really think this is from 40+ years of watching soaps and their behaviour has rubbed off on her

My mother's now dead and I think our relationship may have been better if she gave fewer long phone monologues about people I don't know and I hadn't just switched off and disengaged when she did this.
She used to complain I didn't tell her anything but she rarely gave me a chance to say anything and if I managed a few words she'd then divert on to yet another long story about a random.
It wasn't even "this has happened to X and I'm really upset about it because she's a close friend" which would have given a personal element to it.
I think now I'd have been more assertive and tried to get her more engaged in a 2 way conversation by interrupting her or changing the subject and asking her stuff. Then I just found our phone calls depressing and frustrating which is a shame as we lived several hours apart.

This behaviour has really damaged my relationship with my mum. I avoid her and i feel really sad about it. I’ve spoken to her extensively about it over the years and it’s made no difference at all.

I regularly ask her if she thinks the person she’s gossiping about would be comfortable with what she’s saying.I share very little with her as I know she’ll gossip about me to others. I see less and less of her as there is no real conversation beyond the gossip and mindless details of other peoples lives.

DP's mum does this -

"You know old Fred?"
"Yes, you do - lived down Station Road, next to the chippy"
"You do! He had that dog that used to do that thing with it's feet. His brother was allergic to cheese"
"He sold your dad that car with the squeak"

Ans so on and so on for half an hour until DP cracks and admits that he might know Old Fred just to get to the point ...

"Well, he's dead"

I have a friend who's a bugger for it too, but she's has a bit of a Savior complex and wants to point out where everyone's going wrong. They both do my head in!

I know it seems mean to those who don't understand...but I had bloody years of hearing about Baxters bloody arsehole.

Baxter is the woman who lives downstairs, daughters dog

My Mum will whittle on for ages about people I don't know (while not giving a hoot about us or her grandchildren)

I pointedly ask her now ...do I know these people ? She will still whittle on

It’s quite sad all these people with so little going on in their own lives that they get over enmeshed in the lives of other people. I do tend to stop my older relatives who do this and try and get them discussing current affairs, what they've done, what they've watched on TV. Anything except trivia about people I don't know. I want to hear about them not randoms

🤣🤣🤣

No!
Not at all.

If it's something that I think my family would find interesting, I'll tell them. Otherwise I wouldn't. Why would they care?

My MIL (now deceased) used to do this all the time, it drove us bonkers.
”D’ya remember Breda Kelly?” No, don’t know that name.
”Lives down xxx Road, No. 32” No, I don’t know her.
”Always goes to 10 o’clock mass, sits on the right near the sacristy” No, we don’t go to that mass.
”Little woman, black hair” (grrrr)
”Well, she … blah blah blah”

Haha my retired parents do this, giving me the full blow by blow account of the widowed neighbours love life etc. I’m sure he’d be horrified if he knew how much juicy goss I had on him 😆

I agree with that statement @asrl78

As a side note, having read all the comments, I think people are taking this subject too far the other way!
Of course, myself and others are going to be concerned if friends talk about something serious, either about themselves or friends and family. That goes without saying really.
But I personally do not want to be subjected to the other small, boring ins and outs of their lives.
I am an empathetic person, I just don't see the interest in people I neither really know, nor care about, it's nothing to do with being socially uncivil or unaware!

If the sister is talking about other people then it isn't her life. I wouldn't normally have a problem if it were me on the receiving end, it can be interesting to hear about other people's life experiences that are different from mine, but in this case, if the sister is unloading doom and gloom, I can see how that could be emotionally draining. People have enough things to worry about in their own lives without taking on the problems of people they have no connection with.

My friend, of 30 odd years, constantly does this and it bores the absolute crack off me! She meets up with her old neighbour every few weeks or so (who I have never met and by the sound of her and her weird antics, wouldn't want to!)
I then get the nth degree on everything the friend's said and done since they last met up, none of which interests me in the slightest.
She does it about her family members too (and there are a lot, with a lot of drama!) who I do actually know, but again, have very little interest in the minutiae of their lives.
And the same, if I don't show the emotions I'm supposed to apparently show, I get an indignant look and the story gets repeated again, more dramatically, until I am forced to show some interest to make it all stop! I have tried subtle subject changes, but no, it goes on!
I don't know what is wrong with people that feel the need to do this?! My friend has enough drama in her own life without having to go into every little detail of someone elses shit existence!
I did think it was me being a miserable cow, but I really don't think it is!

Virtuous gossips.

People like this used to be avoided or told to stop gossiping about everyone. So they’ve adapted and worse turned it into a virtue signalling event. Look what a caring person they are worrying about everyone. And look what a cold uncaring person you are for not being concerned as well or attempting to shut it down. There’s examples of it on this thread.

It’s not acceptable to break confidences or to gossip but it is if you’re doing it out of concern. We know it’s gossip when we hear someone gleefully exclaims Omg have you heard Julie’s getting divorced. It looks different when someone relays it in a concerned way but it’s still the same.

This makes so much more sense.

And it means if you don't play the game with them, they can accuse you of being unfeeling rather than having to admit that you've rumbled their ruse.

Your sister is a gossip. Gossiping isn’t socially acceptable so people like your sister present it as if they are concerned. And they need to manipulate you into also being concerned otherwise it’s obvious they are just gossiping.

It doesn't do any good but it's not like I can help it? Like a PP said, some people just get more affected by these things than others. I'm not saying I'm devastated by someone breaking something but I'd be momentarily upset to hear and enquire about how they're doing for a few weeks afterwards. By your logic no one would be upset over anything in the news if they couldn't personally do anything about it. Are people just expressing faux grief over tragic news stories then?

It's showing an interest in her life.
It's not about the friends. It's about her.
You should try to be a bit interested.

I think you've been triggered by this for your own reasons. I don't think there's anything wrong with not caring about people you don't know and there's nothing in my post that says that. It just basically says people have different levels of empathy, not that either is right or wrong.