Research tells me that it might be something called "DTMF TALK OFF" its a widely reported issue. But like Judy, I also am thinking what if some weirdo is out there tapping into my conversations with my friends and family?
Yes, this is it exactly.
The sounds that your phone used to make over the old analog lines were called "DUAL TONE MULTIPLE FREQUENCY" sounds. They were made to be a very specific multi-tone harmonic that dialing equipment could recognize. Before that, they used "pulse" dialing on rotary phones.
DTMF sounds are still in use today for things like menus ("Press 1 to be left on hold, press 2 to be left on hold longer, press 0 to be placed on hold immediately") and the like. The problem is, compressed digital protocols do not render these sounds well enough to be recognized by the remote equipment. Modern phone companies are well aware of this and if they recognize what sounds like a DTMF tone, they will filter it out with software and replace it with a higher-fidelity version of the same DTMF tone in the hopes that the receiving equipment will be able to understand it.
Unfortunately, some people just have "those voices" - my mother is one of them - that occasionally manage to inadvertently mimic a DTMF tone well enough that the phone company tries to reproduce it. It was horrible on Vonage until I managed to convince them to allow me access to the advanced settings of my VoIP adapter, and once I got access to that I was able to turn DTMF detection off - which means any equipment on my premises that depended on DTMF was useless, but I never had any such equipment. Once I went back to a cell line, the problem started right back up.
It has not happened since my mother's land line was upgraded by her local telco to a digital one, probably because the sound artifacts were no longer present to trigger DTMF detection on my end.
You can try asking your carrier to turn off that feature on your line. You may or may not be able to, and it may or may not help since the tones can also be recognized by the carrier of the person you are talking to and "boosted" across to your end.