The best way to get rid of the privet is to paint the stumps with full strenth Brush-B-Gone, but that method works best in summer and early fall. All you can do now is go ahead and cut it to the ground, when it sprouts back up in spring you'll need to spray it with Brush-B-Gone mixed up in a tank sprayer. It may take several times. If you know anyone with a tractor you could always hook a chain to the big ones and pull them up by the roots, that's what my neighbor done. My whole place was infested with privet, and it took me a couple of years to get rid of it.

The thorny vine is most likely Smilax aka Greenbriar. I haven't found anything that kills it yet.

The smilax is probably native but it can survive a cutting (which is good or bad news depending on your viewpoint). Ditto the above comments on privet. You can also try using a Weed Wrench on privets when the ground is not frozen. At any time of year you can at least reduce the biomass by cutting the privets down. To be super careful you can stack up your cut pieces off the ground to avoid the slight potential that the pieces will take root. I did get rid of some smaller privets this summer with foliar spraying. I also uprooted some by hand but it is hit and miss whether any individual plant will come up that way. The Weed Wrench works well but if you have a lot of privets to deal with then spraying the cut stump is more practical. Uprooted plants should not be left on bare soil. One that I did that with knitted itself back into the ground. You will need to do some follow up since those privets probably left some seed and since you can miss the occassional root fragment that regenerates. Still, I think you will see some improvement quickly. Good luck.

Thanks for the advice. If I get a chance, I'll cut it down. I have haul away service from the city so they won't have a chance to root laying on the asphalt by the curb. I may be able to get my truck in there or a come-along to pull out the smaller ones. Some have been here for more than 15 years according to neighbors. If they regrow, at least they won't look so ragged.

Amazingly, deer will eat the tender resprouts of privet. Several years ago we attempted pulling out and cutting, and that winter noticed a herd (five) of deer nibbling on the new growth. So, if you are not able to spray at this moment, maybe the deer will prove useful and help keep the privet trimmed!

I notice in that photo it looks like it slopes down to a wetter area. I know first hand that privet loves water, and I rarely see it in areas where it gets very dry. There is a stream down the road from my home and both sides have a solid wall of privet.

Yes. It does slope down and the privet is growing in what used to be a stream. At some point, the city diverted the stream and now only runoff from two street drains fills that space. It's always wet though. There is also English Ivy covering the ground. I have no urge to fight that battle. LOL

Privet seems to like all kinds of terrain around here. You can cut the English Ivy where it has gone up trees and leave the cut portions on the tree to wither and die. English Ivy usually pulls out of the ground pretty easily as well but it is time consuming.