A sharing person is generous and willing to share. If you are sharing, you are glad to give things away to others. So can we borrow five dollars?
To share something is to use it or enjoy it with others, like when people share a cake at a birthday party. The adjective sharing has a related meaning, but the focus is more on unselfishness. A sharing person might try to divide up not just the cake, but also all the food in his refrigerator — just to be sure everyone is fed.