No, that’s not Privet. Perhaps Elder??? Any buds visible?

Not privet. I'm thinking more along the lines of Butterfly Bush, Buddleia for the second picture and yes to Hawthorne which is native for the first.

Effective techniques of removal will vary by the plant.

The second photo gives off vibes of one of the invasive honeysuckles. Check to see if the stems are hollow. Most, if not all of the invasive lonicera have hollow stems. Native honeysucle will have solid stems. The bark often will be shaggy and peeling on older stems. Look at the twigs and buds. The buds of lonicera will be opposite and almost at right angles to the stem.

Of course the ID would be easier with leaves and blooms, lol.

Currently, there is about two feet of snow keeping me from most of this, but we're supposed to get a warm-up this week, so I'll see if I can get to them this week and see if any buds have started to form.

I wasn't aware that we had any native hawthorns!

I'll have to check to be sure, but I do believe that these stems have a hollow core.

The hawthorn could be a native or non-native. You would need a correct species ID. They have wicked thorns so I remove them.

The pictured Hawthorn isn’t Crataegus monogyna which is invasive in the US. Whether you have other invasive Hawthorns I don’t know.

Crataegus is also a genus with several hundred species that are very likely promiscuous and highly variable making a definitive ID more difficult. Except for the thorns, they can be quite lovely and make good deer food.

The first photo looks like Dotted Hawthorn (C. punctata), but I say that only because the leaf is similar and it is common in the eastern U.S. In any case, I suspect this tree is native. English Hawthorn (C. monogyna) is classified as invasive in California, but not in the east.

The second photo looks like Amur Honeysuckle. I can't speak to New York, but in the upper midwest it is highly invasive. I've removed a lot of it. If this is what you've got, you'll want a plan to remove it ASAP.

You're on the right track to identify plants before removing them. This will be much easier once you can see fresh foliage, flowers, and fruit.

Question for Christopher: Do you remove all plants with thorns and why?

Pretty much. I am a working gardener and I hate thorns. There are so many plant choices in this world there is simply no reason to be getting stabbed all the time.

How that works out in real life is another matter. Thorns are hard to avoid in the plant world. In my garden they are confined to tenacious weeds like Blackberry and Greenbriar/Smilax.