We have shrubs (bushes) in the front of our house that I had previously identified as Chinese privet (photo 1 & LOWER RIGHT of photo 3), but now I'm not sure. We keep them trimmed for the most part, so I never even knew that they would flower until this past year; then we lopped off the flowers. The one in the photo had been trimmed down to a 1-foot tall stem/trunk only, and bushed back out this fluffy (you're seeing it about 2.5 feet tall and 2 feet wide in that photo).
In the back of our house, we have shrub-trees (photos 2, 4 and 5 plus TOP LEFT of photo 3). When I did a google image search for their clusters of purple berries, the only things I found that were even similar were ALSO listed as privet... but these are NOT the same thing as the bushes in our front yard.
Front yard: 1-3" long leathery leaves grow across from each other, all the way up every branch, even branches that have other branches coming from them. Glossy, bright medium green, pointed oval shape just slightly larger at the stem end. Very smooth, veins don't make any texture on the front and just barely show through on the back. The edges of the leaves tend to be thick/hard and curl under ever so slightly. When the flowers appeared, they looked like white lilac flowers (came out looking like little fists and opened into little four-petal crosses). The stems have white freckles/bumps on new growth. When the plants are trimmed, they grow in very bushy.
Back yard: Leathery, semi-gloss, medium-to-dark green oval leaves have visible vein grooves on front AND back (ie, it's an indent on the front AND an indent on the back.) They are mainly about 2-3" but can be as large as 4" long and have a pinch-point end. They grow in clusters, not opposite one another, and there are some branches that don't have leaves growing on them. The new growth on the stems has white freckle spots. They flower in the spring/summer (sadly I can't remember what shape/color), and those turn into matte, dark purple grape-like clusters of baby-pea-sized berries. They have that dusty-looking coating like blueberries have. This plant has grown 30 feet tall. The stems/trunks range from 3" to about 8" in diameter when it grows tall (multiple trunks can fuse into a larger size than that, but I'm talking about single "stems" that grew into a trunk.) Drastically trimming these plants results in having new long "sucker" stems (seen in photo 2 growing near the ground, and also see bottom center of photo 5, bushier versions growing from a cut branch wound that is about 4' above the ground).
The reason I'm asking is because we want to get rid of the ones in the back. No matter how much we cut it back, it will regrow - there have been years that we cut one or more of the trunks down to 2-4 feet and no branches at all, and within a single year they shoot back up to match their neighbors. We can't afford to have someone come grind the stumps, so I'm hoping I can ID it and figure out the best way to kill it after chopping it down.