It depends on your jurisdiction. In my state, you can trim whatever hangs over your property line --- at your cost.
If the tree is a hazard because it has dead limbs or any other problems, you can notify your neighbors that they need to do something about it.
However, some people may actually appreciate having a large tree next to their property, needles and all.
With respect to the backyard, are there any ordinances about junk and not mowing? If not, just build the fence high enough.

Have you always been civil with this neighbor? My sister's yard was an eye sore to a next door neighbor. The neighbor politely asked if my sister could do something. She wound up building a trellis to block the view.
The neighbor's house sold after that. My sister does keep the front and any property that is in public view neat and maintained.
Maybe they will split the cost. After all, this tree could very well fall on their property.

Call your insurance company and tell them you are worried about the tree and the damage it could cause if it fell.

They will call the neighbor's insurance company and let them know there is a hazard and ask them to see about "mitigating" that hazard.

We did that with some HUGE pines that swayed in a breeze and were big enough to land on our bedroom if they fell ... and the two insurance companies split the cost of the removal, and I let the arborists use our yard for access to keep the cost down.

I wouldn't call my insurance company, & I sure wouldn't count on them calling anyone else's insurance company.

Chances are your company would just non-renew your policy due to an increased risk or hazard, & while I'm happy that lazygardens' carrier & her neighbor's carrier paid for removing the trees, I myself cannot imagine any company these days paying for such a thing.

Maintenance of the property & removal of hazards is the homeowner's responsibility, not the carrier's.

The carrier pays for loss due to covered peril, not for maintenance or loss prevention.

In your shoes, to keep things pleasant between us, I might tell my neighbors that my carrier had non-renewed pending my getting the branches removed, & then I'd get the branches removed.

Nothing seems to suggest that the tree in question is a hazard, or is it?

"If the tree is a hazard because it has dead limbs or any other problems, you can notify your neighbors that they need to do something about it."

We just had an issue like this. We regret contacting the neighbors. If we had it to do over again we would have called the city and asked them to notify the neighbor of the nuisance trees. Like our other neighbor did. This is why.

We had several dead trees (60ft+) looming over our house. We let the neighbor know. We asked him to please let us know what he planned to do and when he planned to do it. We said his workers could use our yard to access the trees but that we wanted to be home when it happened.

Fast forward a month. We went out to lunch and discovered tree workers in our back yard when we got home. I asked who they were and what they were doing on my property. They said they worked for the neighbor and didn't have to tell me who they were. I said "You are on MY property. I have the right to know WHO is on MY property. If you do not want to tell me this, (or show me proof that you are registered to do tree work in this community) then I will call the police and have you arrested for trespassing."

The guy said he wasn't going to tell me so I called the police. In the meantime he yelled to his guys "She is crazy, she's calling the police, hurry up and get these logs out of here"

I grabbed my camera and videotaped him and his workers rushing to drag the logs over to the neighbors property and wrote down the license plate number of his truck. The police showed up and told them to stay off our property.

Then the neighbor showed up. I said, "We asked you nicely to tell us when you planned to have the trees removed. You ignored us. Not only that, but you deliberately arranged to have the trees removed at a time when you knew we wouldn't be home. So much for 'cooperation'.

Do not come on my property for any reason, ever again or we report you to the police and sue you.

The following monday we went to the city/zoning dept. I showed them the pics/video and explained what had happened. They said tree workers in our city are required to be registered/bonded. We were told to call if we saw them again. They said they would be happy to send out an inspector to check if they were licensed and cite them if they weren't.

They (zoning dept) were also nice enough to share information about another neighbor adjacent to this neighbor who had contacted them about the same problem. Instead of doing it the way we did (what we thought was the 'nice way'), they went straight to the city.

The zoning dept/city sent our neighbor a notice to have the trees removed. He could do it in a timely manner or they would do it for him and send the bill.

Sometimes it pays to be nice. Other times not. Why not see what your city says before you talk to your neighbor? You will be in a better position to know your rights before your neighbor chooses to ask for forgiveness, rather than...

Licensed vs. unlicensed tree work can be a real problem. It isn't my place to tell a neighbor to spend thousands of dollars more, which they may not have, to hire a licensed tree company for their property. But if the crew has to work from my yard, then it is my business.

No one respects anyone's property rights anymore. So, you are leaving the neighborhood, and before you go you want to permanently alter the neighbors property and possibly significantly reduce it's value? (Trees increase the value of a piece of property). All because potential buyers *MIGHT* have an issue with the trees? Did any realter or appraiser say the branches would be a problem? Did an arborist say the tree might be a risk?

If I were your neighbors and I got wind you were calling the insurance company, I'd make a point of finding out when you were showing the property and throwing loud parties.

Let it go. Pine needles are a part of life and nature. Prospective buyers are as likely to like the shade and wooded look as dislike the needles. It's not worth the risk you will alienate your neighbors. Is there anything about the neighborhood you'd prefer potential buyers didn't know? You are particularly vulnerable and are about to make a nuisance of yourself by having lots of strangers wander through the area, having moving trucks come by, etc.