I just started to deal with this, as my son loves to put a huge amount of paste on his brush. I am teaching him it should be a TINY amount. He then swishes and spits with water and tries to take off any paste in the sink (before it dries). Soon, he'll have to clean the sink after he brushes, but we're not there yet. He'll be reward (sticker or such) when he does a good job.
I don't expect too much as his father, still leaves a big mess in his sink and uses half a tube with each brushing (lo)!