Jesus did not answer their demands by giving them a sign immediately but foretold about His own death and resurrection that will be the ultimate sign of His authority given to everyone.
Everyone knows John 3:16 by heart but the next verse is equally important, for Jesus did not come to condemn us but save us. If he wanted to condemn us for our wrongdoings, He would have brought the day of judgment upon us but rather He came with the sole intention to be an atonement for our sins.
The living water represents His Spirit in us who gives us the joy that comes from knowing that our sins are forgiven. The joy in our hearts that gives us a new life, a new beginning with the knowledge that the slate has been wiped clean. That is what Jesus has to offer.
Quite ironical that the Jews , through whom the salvation was to come rejected the Messiah but the ‘pagan’ Samaritans were the first to proclaim Jesus as the Savior of the world. Jesus revealed His true identity in this Samaritan town, something that He would not do in the Jewish towns.