Normal practice (in a domestic garden, that is) is to prune the branches quite close to ground level. Dogwood twigs are produced on the stem in opposing pairs, so if you cut just above a pair of buds, two more branches will grow - like this:
So after a few seasons you can get lots of stems from each plant.
In retrospect I think I should have cut those branches shorter - maybe six inches or so above ground level.
In one corner of my garden I have adopted a different pruning strategy for a clump of Dogwoods. I have cut them at about 4 feet above ground level. They are therefore slim down below, with twiggy tops - almost like a "standard" shape:
The green leaves belong to a Clematis plant which has become entangled with the Dogwoods
I have kept the upward-pointing twigs, and removed most of the downward- or sideways-pointing ones.
What this means is that the plants growing at the base of these shrubs (currently Euphorbia and Primroses) get a decent amount of light, whilst I still get a host of brightly-coloured Dogwood stems.
Euphorbia at the feet of the Dogwood plants
Best practice suggests that you should prune your Dogwoods very severely in order to keep them vigorous, which in turn tends to produce stems with brighter colours. I usually trim mine quite gently each Spring to keep them from getting too big, and then give them a severe chop every 3 years or so. It seems to work for me.
