Those are elaeagnus shrubs! I’m not sure what to tell you about the shoots, but maybe you can research a little now that you have the name. My neighbors have mature ones and they’re so pretty. I’ve never noticed the shoots, but I’m sure theirs’ gets them on occasion too because it’s the nature of the plant.

Elaeagnus is often recommended as a good shrub, but probably only if you let that particular variety grow natural. I've noticed how in my neighbor's yard they throw out those long stems after trimming. Weird. I agree with Lindsay that it seems to be their natural propensity to do that and there's no way to control it.