I work for a major phone company here in Canada and started as an operator. I'm not terribly familiar with mobile phones, but there is a lot about landlines that people aren't aware is possible.
Lines get crossed all the time, which can result in hearing other people's conversations. Sometimes the communication can be one way or two way. People will often call the operator to tell us that the lines are being tapped or other paranoia.
A relatively common thing that we were able to do (assuming provincial law allowed it), is listen into lines. It's intentionally distorted, but a trained ear can hear what is being said. People will ask for is to do this when they've been trying to get in touch with someone for a while and the line has been busy. 90% of the time it is clear that the phone is off the receiver, but sometimes there will be people on the line. The next step, if laws allow for it, would be to break into the line. When an operator does this, a loud screech sounds for all 3 people (including the operator) on the line, usually followed by one or two "what the hell was that?". Then the operator introduces themselves and let's the party know that someone is trying to get a hold of them. I've never witnessed this being abused, but one has the potential to really confuse people.
Another duty as an operator is relay service for the deaf. Generally, this is pretty straight forward and the operator will speak on behalf of the deaf person who is typing a message, then type the response back to the deaf person. Much of the time, people immediately assume that these calls are strange, or a prank, if they weren't expecting the call. Try ordering a pizza like this. The other catch is that the operator is not allowed to speak to the hearing party, and can only speak what is being typed. Some deaf people will directly translate from sign language, which isn't constructed the same way as English, or there will be errors in the typed word that simply occur as a result of the technology. To someone who is not familiar with this, it can be a very bizarre experience.
And then there are VOIP services that just mess everything up. For the example above, you can cut the operator out of the equation and have a robot speak the typed word. The deaf user does not have any obligation to explain what is occurring and the technology has the potential to be abused by anyone at all.
Also on VOIP, you can create spoofed numbers. People will generally be quite spooked to receive a call from their own number.
Hopefully this info helps debunk some of the strangeness that people experience.
Not one to let people down, I'll finish the post with a story of my own, while working a graveyard shift... (Be warned, it's not actually that interesting)
I was working a graveyard shift one night. Generally these are pretty boring and there would be maybe 2-3 people working. One might go on break and leave the other alone in the building. It's a pretty old building, 3 floors of desks with large windows to the front and the back. Each floor is one large room where you can see everything if you stand up. For a graveyard shift, we are restricted to the 2nd floor. The first floor doesn't house any equipment and the 3rd is locked (the 3rd floor also does not house any operator equipment). There is a lone security guard who sits in the cafeteria on the first floor, but the building is empty otherwise.
We are used to getting all sorts of prank calls and perverts calling in at these times, and having to relay (again, for the deaf who are typing) the occasional sex chat line. Otherwise, it is common to go for 30 minutes without receiving a call. Maybe even an hour, if it was a slow night.
I wasn't alone, but my coworker on this night is a friend of mine who was sitting beside me watching movies or something. I answered a call, and nobody was on the other line, except for my own voice giving the same spiel that I always gave when answering a call. I heard my own voice saying something like:
"This is the operator. How can I help you? [delay] Operator. How can I..."
My voice was cut off at that point, but the line was still active. I could hear the empty air, but you get a feel for these things, and I could feel that there was someone listening on the other line. I got spooked and disconnected.
Now, the computer screen shows the number that is calling, and this time it was completely blank. Sometimes it will be blank, but it is due to a software glitch, or 9-11 calling, and it doesn't happen often. Also, keep in mind that only landlines can call the operator, so this wasn't a VOIP or mobile. The equipment that we used is also the same equipment that was used in the 70s, very primitive, and did not have any way of recording audio.
I'm sure that there is a reasonable explanation for the occurrence and that it was likely a prankster who somehow knew that they had a 50/50 shot of getting me again... But the whole thing was enough to creep me out that night. It was business as usual after that- no second occurrence and my coworker didn't receive anything either.