Your best bet is talk to your neighbor, but don't go with a chip on your shoulder.
I had a similar situation only with a half dozen white pine. I didn't want the trees gone, just wanted to trim the bottoms up for a better air flow since my shed is directly underneath. I mentioned taking the height up to 15 feet. Not only did I get the okay, but he was pleased that I thought to ask.
On the other side of the coin, I had a friend who had a tree topped during a wind storm which took out part of a chainlink fence at the back of the property. The part of the tree that remained was leaning heavily over her neighbors shed and if it were to fall would take out the shed and the chainlink fence it was next to.
I was over at her house partaking in some really good bbq and I mentioned the tree. She in turn voiced her concern but also mentioned it would cost too much to hire someone to come in and remove the tree. My return comment was I would be back later in the week to fell the remainder of the tree (still a good 40 feet tall) and left it at that.
I guess most people tend to just make offers and never follow through because my friend didn't expect me to do what I did. I showed up as I said, armed with a chainsaw, b&c oil, fuel mix, felling wedges, and a rope to help guide the tree where I needed it to go. Anyway, my friend was asleep at the time, but I just went ahead and tied off the tree and felled it back into her yard where I proceeded to limb n buck it. When I was through, and in the process of stacking all the wood, she comes out in her night cloths still wiping the sleep from her eyes. She was surprised I had followed through.
Her neighbor who had been watching told her she should have watched because "that is the way a professional fells a tree." Well, truth be told, I am not a professional but I do have plenty of experience. Total cost for her $0! She still thanks me every time they have a wind or ice storm for what i did. Anyway, my point is, it is entirely possible your neighbor is just as concerned about the tree as you are but likely doesn't have the skill or tools required to take it down.
If that's the case, either split the cost of the removal or, if you have the skills, take the tree down with your neighbors consent and if you have a wood stove, IWB, or OWB, take the wood for that.