As a elderly widow, who 3 years ago, my (late) husband had dementia. At that time, with my husbands medical bills and other expenses in my big house that was 50 years old, we could barely afford to just paying monthly expenses, since due to the age of the house, we had problems with the AC or the furnace or other problems due to the age of the house. A fence if we had had one and if damaged, then that is (not) necessary to fix.
I have since moved and sold the big house and paid cash for a much smaller 2 year old house, thus my living expenses are about 1/3 of my prior old house. Now with my new house, I installed a fence for my dog and had to have a survey done (cost $500) and my HOA had to approve it, and if not done right, then they could remove it at (my) cost.
If, I had to guess, your neighbors being elderly and with the husband having dementia is probably living on SS income (only) and are barely making expenses, just like me 3 years ago, thus the reason for the lady being mean, since my guess is that they (can't) afford even 1/2.
You and another neighbor have an advantage of being able to use 1 side each to help keep your dogs safe. The other neighbor has done minor things to fix, but you want them to pay 1/2 instead?.
Now, unless you live in an HOA, then the HOA could have the (entire) fence taken down. Or if they have violated a code violation, or if you have a survey and find out that their fence is in (your) property. But like others have suggested, build a fence on the side of your property, then any damages to their fence is their problem, not yours.