Do gutters have to have downspouts?

We bought a house a couple months ago and just had our roof re-shingled. We'd noticed that there were not gutters in the back of the house and the water was starting to wear the deck. We found 3 sections of gutter and brackets that had obviously been there previously. We had our roofer put them up to protect the deck but there are no downspouts. They capped the one end so the water all runs to the other end. Do we have to put a downspout on that end or can we use a splash block to prevent a hole being worn in the flower bed. Without a downspout, are we opening ourselves up to possible water problems in the basement? The water doesn't drop from the gutter right next to the house; there is some space. Being a first time home owner, I know nothing about gutters!