Depends. How much rain do you get at any one time? Is a normal rain just enough to just moisten the material ion your compost pile or does it saturate that material and require you to repile them?
The last bins I built were 4 x 4 so half a sheet of
T-111 siding could be placed on as a cover to aid in keeping both rain and snow off and that allows me to better control the moisture level in the material. Many people do not cover their compost piles and do not have much of a problem with that, but plans for several compost bins do include covers.
Many of the compost bins I saw in Ireland and England were covered while few I have seen on the great plains here are. There is no one answer to this question.

I tend to cover or not depending upon the weather. If the compost is dry and it's going to rain, I leave covers off. I usually try to cover them in winter so they don't get leached out by all the rain and snow when they're too cold to do much decomposing anyway.

Remember that whatever leaches out ends up in the soil beneath, so if you move your composting location now and then, you'll improve different areas. I have several raised garden beds, and one is usually fallow with a compost pile built on it. I do a different one every season. It's really great for the soil beneath, especially if it's poor to begin with.