You can take cuttings from Cotoneaster plants and it could even benefit the parent plant as a pruning (good city sell point for chopping off a few branches from an existing plant?). Cuttings from an existing plant in the same area will be sure to work out since those plants were already doing well and it's just a clone.
Here's what I found on the web about em, either technique works.
Softwood cuttings are prepared from soft, succulent, new growth of woody plants, just as it begins to harden (mature). Shoots are suitable for making softwood cuttings when they can be snapped easily when bent and when they still have a gradation of leaf size (oldest leaves are mature while newest leaves are still small). For most woody plants, this stage occurs in May, June, or July. The soft shoots are quite tender, and extra care must be taken to keep them from drying out. The extra effort pays off, because they root quickly.
Semi-hardwood cuttings are usually prepared from partially mature wood of the current season’s growth, just after a flush of growth. This type of cutting normally is made from mid-July to early fall. The wood is reasonably firm and the leaves of mature size. Many broadleaf evergreen shrubs and some conifers are propagated by this method.
here's the resource: