I am not an electrician, but I've been researching a lot recently to try and figure out why my system is behaving so strangely.
Main problem: Lights flickering, sometimes quite severely, and appliances sometimes just turning themselves off or not working properly (e.g. microwave seemingly working but not generating much heat). Last night, for example, my AppleTV turned itself off multiple times while watching a one hour show.
More Info: I have two meters, which feed into two panels that are next to each other, one for the basement, and one for the 2nd floor. The 2nd floor panel feeds into a sub-panel next to it which is for the 1st floor.
I bought some electric meters and testers and have found some distressing things. With a Kill-A-Watt style plug-in meter, I've found that the voltages in my house are often not great. If I try to turn off everything in the house, the voltage everywhere is generally around 120V +- 3V. However, if I put a significant load on (like a space heater or bathroom fan heater), the voltage in half the outlets in the house goes down to around 95V while the other half of the outlets go to 135V. These voltages are not just a transitory spike, they persist as long as I keep the load on. And this is not a enormous, unreasonable load I'm putting on; my space heater that I have been using to make the voltage drop to 95V draws something like 9 amps. I also find that if I pile on more load, I can get one side as low as 75V while the other side goes up to 145V persistently.
I mapped all my lights, outlets, and appliances to their circuit breakers, and I found that all the outlets that go down in voltage together are on one leg of my service, whereas all the outlets that go up are on the other leg of my service. The leg that goes down in voltage is the one where I put the extra load, and I can change which leg goes down and which one goes up in voltage by plugging my space heater into outlets of different legs. The lights that flicker or dim are always on the service leg that has low voltage.
The other interesting thing is that I used a clamp meter on my ground wires from the panel, and I found a significant amount of current flowing through the ground wires between the panels and out to the water main. If I put the space heater in the basement, for example, I can get as much as 5 amps flowing on the ground wires out to the water main, and the clamp meter shows that 5 amps is flowing on the pipe itself out to the outside world. I'm far from an expert, but this concerns me.
I also bought an outlet tester, and I tested almost all the outlets in my house (There are four I didn't test because I didn't want to invade my houseguests' privacy in their bedroom). All but two outlets tested fine, and those two showed reversed hot/neutral.
Several electricians have come in and worked on the system. Each have found real problems and fixed them, and things sometimes get better for a while, but often return. The power company came months ago and found the neutral wire basically totally disconnected in the power feed and the meters. An electrician tightened up a bunch of neutral wires in the panel. Later, he installed the ground system to the water main (there was no connection to the water main before). Another electrican just this week disconnected the ground from the neutral in the sub-panel.
I have another electrician coming in to diagnose and work on the problem tomorrow, but I'm curious if anyone has good ideas as to what might be causing this problem and/or what sorts of tests I should be doing to narrow down the culprit. Thanks!
UPDATE: A power company worker came out and opened up the service box where my meters are connected to the outside. He found that while both hot wires had current flowing, there were exactly 0 amps on my neutral. He undid the neutral connection and replaced it but the neutral still read 0 amps. He agrees the problem is on their side somewhere. Now he's sitting in his van watching TV on his phone while he waits for something. I'm not sure what.
FINAL UPDATE: It took about 4+ hours, but they got it fixed, and I have rock solid 120v no matter what load I put on the house. I'm over the moon. They ended up bringing in a tanker truck with a giant vacuum and pressure washing out the maintenance hole, which was literally completely full of gunk. Eventually they uncovered the wires underneath the muck, connected the neutral, and it's all fantastic. Thanks so much to folks here, especially @Harper - Reinstate Monica, who gave a phenomenal response that was very helpful.