Both are possible, but the meaning changes totally! It depends on what the subject of the sentence is.
I need to drink = I'm thirsty and I have to drink something
This milk needs drinking = This milk needs to be drunk
You'd say the second one if a carton of milk has been in the fridge for a few days, and you are telling someone that the milk should be drunk before you open a fresh carton, for example.
I need to wash = I must take a shower
The car needs washing = Someone needs to wash the car.
I hope that's clear.